Calculus Examples

$\frac{\sec (x) \tan (x) + \sec^{2} (x)}{\sec (x) + \tan (x)}$

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sec (x) \tan (x) + \sec^{2} (x)$ and $g (x) = \sec (x) + \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) \frac{d}{d x} [\sec (x) \tan (x) + \sec^{2} (x)] - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

By the Sum Rule, the derivative of $\sec (x) \tan (x) + \sec^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [\sec (x) \tan (x)] + \frac{d}{d x} [\sec^{2} (x)]$ .

$\frac{(\sec (x) + \tan (x)) (\frac{d}{d x} [\sec (x) \tan (x)] + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec (x)$ and $g (x) = \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Multiply $\sec (x)$ by $\sec^{2} (x)$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{1} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(\sec (x) + \tan (x)) ({\sec (x)}^{1 + 2} + \tan (x) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $2$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan (x) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan (x) (\sec (x) \tan (x)) + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\tan (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x) + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\tan (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x) + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x) + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + \frac{d}{d x} [\sec^{2} (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (x)$ .

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To apply the Chain Rule, set $u$ as $\sec (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + \frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 u \frac{d}{d x} [\sec (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Replace all occurrences of $u$ with $\sec (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec (x) \frac{d}{d x} [\sec (x)]) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec (x) (\sec (x) \tan (x))) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\sec (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 (\sec^{1} (x) \sec (x)) \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\sec (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 (\sec^{1} (x) \sec^{1} (x)) \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 {\sec (x)}^{1 + 1} \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Differentiate using the Sum Rule.

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Add $1$ and $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

By the Sum Rule, the derivative of $\sec (x) + \tan (x)$ with respect to $x$ is $\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)]$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - (\sec (x) \tan (x) + \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\sec (x) \tan (x) + \sec^{2} (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - ({(\sec (x) \tan (x) + \sec^{2} (x))}^{1} (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\sec (x) \tan (x) + \sec^{2} (x)$ to the power of $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - ({(\sec (x) \tan (x) + \sec^{2} (x))}^{1} {(\sec (x) \tan (x) + \sec^{2} (x))}^{1})}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{1 + 1}}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $1$ .

$\frac{(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Simplify.

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Simplify the numerator.

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Simplify each term.

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Expand $(\sec (x) + \tan (x)) (\sec^{3} (x) + \tan^{2} (x) \sec (x) + 2 \sec^{2} (x) \tan (x))$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\sec (x) \sec^{3} (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Simplify each term.

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Multiply $\sec (x)$ by $\sec^{3} (x)$ by adding the exponents.

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Multiply $\sec (x)$ by $\sec^{3} (x)$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{1} (x) \sec^{3} (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sec (x)}^{1 + 3} + \sec (x) (\tan^{2} (x) \sec (x)) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $3$ .

$\frac{\sec^{4} (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec (x) (\tan^{2} (x) \sec (x))$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + \sec^{1} (x) \sec (x) \tan^{2} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + \sec^{1} (x) \sec^{1} (x) \tan^{2} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + {\sec (x)}^{1 + 1} \tan^{2} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec (x) \sec^{2} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Move $\sec^{2} (x)$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 (\sec^{2} (x) \sec (x)) \tan (x) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec^{2} (x)$ by $\sec (x)$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 (\sec^{2} (x) \sec^{1} (x)) \tan (x) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 {\sec (x)}^{2 + 1} \tan (x) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $2$ and $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan (x) (\tan^{2} (x) \sec (x)) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\tan (x)$ by $\tan^{2} (x)$ by adding the exponents.

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Move $\tan^{2} (x)$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{2} (x) \tan (x) \sec (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\tan^{2} (x)$ by $\tan (x)$ .

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Raise $\tan (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{2} (x) \tan^{1} (x) \sec (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + {\tan (x)}^{2 + 1} \sec (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $2$ and $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + 2 \tan (x) \sec^{2} (x) \tan (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $2 \tan (x) \sec^{2} (x) \tan (x)$ .

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Raise $\tan (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + 2 (\tan^{1} (x) \tan (x)) \sec^{2} (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\tan (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + 2 (\tan^{1} (x) \tan^{1} (x)) \sec^{2} (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + 2 {\tan (x)}^{1 + 1} \sec^{2} (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $1$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + 2 \tan^{2} (x) \sec^{2} (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Reorder the factors of $2 \tan^{2} (x) \sec^{2} (x)$ .

$\frac{\sec^{4} (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) + 2 \sec^{2} (x) \tan^{2} (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $\sec^{2} (x) \tan^{2} (x)$ and $2 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \tan (x) \sec^{3} (x) + \tan^{3} (x) \sec (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Reorder the factors of $\tan (x) \sec^{3} (x)$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Add $2 \sec^{3} (x) \tan (x)$ and $\sec^{3} (x) \tan (x)$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - {(\sec (x) \tan (x) + \sec^{2} (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Rewrite ${(\sec (x) \tan (x) + \sec^{2} (x))}^{2}$ as $(\sec (x) \tan (x) + \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x))$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - ((\sec (x) \tan (x) + \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Expand $(\sec (x) \tan (x) + \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x))$ using the FOIL Method.

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Apply the distributive property.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec (x) \tan (x) (\sec (x) \tan (x) + \sec^{2} (x)) + \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Apply the distributive property.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec (x) \tan (x) (\sec (x) \tan (x)) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Apply the distributive property.

Simplify and combine like terms.

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Simplify each term.

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Multiply $\sec (x) \tan (x) (\sec (x) \tan (x))$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{1} (x) \sec (x) \tan (x) \tan (x) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{1} (x) \sec^{1} (x) \tan (x) \tan (x) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - ({\sec (x)}^{1 + 1} \tan (x) \tan (x) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan (x) \tan (x) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\tan (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) (\tan^{1} (x) \tan (x)) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Raise $\tan (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) (\tan^{1} (x) \tan^{1} (x)) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) {\tan (x)}^{1 + 1} + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec (x) \tan (x) \sec^{2} (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Move $\sec^{2} (x)$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{2} (x) \sec (x) \tan (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec^{2} (x)$ by $\sec (x)$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{2} (x) \sec^{1} (x) \tan (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + {\sec (x)}^{2 + 1} \tan (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Add $2$ and $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \sec^{2} (x) (\sec (x) \tan (x)) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec^{2} (x)$ by $\sec (x)$ by adding the exponents.

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Move $\sec (x)$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \sec (x) \sec^{2} (x) \tan (x) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec (x)$ by $\sec^{2} (x)$ .

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Raise $\sec (x)$ to the power of $1$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec^{2} (x) \tan (x) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + {\sec (x)}^{1 + 2} \tan (x) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Add $1$ and $2$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \sec^{3} (x) \tan (x) + \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec^{2} (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \sec^{3} (x) \tan (x) + {\sec (x)}^{2 + 2})}{{(\sec (x) + \tan (x))}^{2}}$

Add $2$ and $2$ .

Add $\sec^{3} (x) \tan (x)$ and $\sec^{3} (x) \tan (x)$ .

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Apply the distributive property.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - (\sec^{2} (x) \tan^{2} (x)) - (2 \sec^{3} (x) \tan (x)) - \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Simplify.

$\frac{\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x) - \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Combine the opposite terms in $\sec^{4} (x) + 3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x) - \sec^{4} (x)$ .

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Subtract $\sec^{4} (x)$ from $\sec^{4} (x)$ .

$\frac{3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x) + 0}{{(\sec (x) + \tan (x))}^{2}}$

Add $3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x)$ and $0$ .

$\frac{3 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x)}{{(\sec (x) + \tan (x))}^{2}}$

Subtract $\sec^{2} (x) \tan^{2} (x)$ from $3 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{2 \sec^{2} (x) \tan^{2} (x) + 3 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x) - 2 \sec^{3} (x) \tan (x)}{{(\sec (x) + \tan (x))}^{2}}$

Subtract $2 \sec^{3} (x) \tan (x)$ from $3 \sec^{3} (x) \tan (x)$ .

$\frac{2 \sec^{2} (x) \tan^{2} (x) + 1 \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Multiply $\sec^{3} (x)$ by $1$ .

$\frac{2 \sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Simplify the numerator.

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Factor $\sec (x) \tan (x)$ out of $2 \sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x)$ .

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Factor $\sec (x) \tan (x)$ out of $2 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{\sec (x) \tan (x) (2 \sec (x) \tan (x)) + \sec^{3} (x) \tan (x) + \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Factor $\sec (x) \tan (x)$ out of $\sec^{3} (x) \tan (x)$ .

$\frac{\sec (x) \tan (x) (2 \sec (x) \tan (x)) + \sec (x) \tan (x) (\sec^{2} (x)) + \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Factor $\sec (x) \tan (x)$ out of $\tan^{3} (x) \sec (x)$ .

$\frac{\sec (x) \tan (x) (2 \sec (x) \tan (x)) + \sec (x) \tan (x) (\sec^{2} (x)) + \sec (x) \tan (x) (\tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Factor $\sec (x) \tan (x)$ out of $\sec (x) \tan (x) (2 \sec (x) \tan (x)) + \sec (x) \tan (x) (\sec^{2} (x))$ .

$\frac{\sec (x) \tan (x) (2 \sec (x) \tan (x) + \sec^{2} (x)) + \sec (x) \tan (x) (\tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Factor $\sec (x) \tan (x)$ out of $\sec (x) \tan (x) (2 \sec (x) \tan (x) + \sec^{2} (x)) + \sec (x) \tan (x) (\tan^{2} (x))$ .

$\frac{\sec (x) \tan (x) (2 \sec (x) \tan (x) + \sec^{2} (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Factor using the perfect square rule.

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Rearrange terms.

$\frac{\sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sec (x) \tan (x) = 2 \cdot \sec (x) \cdot \tan (x)$

Rewrite the polynomial.

$\frac{\sec (x) \tan (x) (\sec^{2} (x) + 2 \cdot \sec (x) \cdot \tan (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \sec (x)$ and $b = \tan (x)$ .

$\frac{\sec (x) \tan (x) {(\sec (x) + \tan (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Cancel the common factor of ${(\sec (x) + \tan (x))}^{2}$ .

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Cancel the common factor.

$\frac{\sec (x) \tan (x) {(\sec (x) + \tan (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Divide $\sec (x) \tan (x)$ by $1$ .

$\sec (x) \tan (x)$