Calculus Examples

$lim_{x \to 0} \frac{\ln (\tan (45 + a x))}{\sin (b x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} \ln (\tan (45 + a x))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 0} \tan (45 + a x))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.1.2

Move the limit inside the trig function because tangent is continuous.

$\frac{\ln (\tan (lim_{x \to 0} 45 + a x))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\ln (\tan (lim_{x \to 0} 45 + lim_{x \to 0} a x))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.1.4

Evaluate the limit of $45$ which is constant as $x$ approaches $0$ .

$\frac{\ln (\tan (45 + lim_{x \to 0} a x))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.1.5

Move the term $a$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (\tan (45 + a lim_{x \to 0} x))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\ln (\tan (45 + a \cdot 0))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $a$ by $0$ .

$\frac{\ln (\tan (45 + 0))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.3.2

Add $45$ and $0$ .

$\frac{\ln (\tan (45))}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.3.3

The exact value of $\tan (45)$ is $1$ .

$\frac{\ln (1)}{lim_{x \to 0} \sin (b x)}$

Step 1.1.2.3.4

The natural logarithm of $1$ is $0$ .

$\frac{0}{lim_{x \to 0} \sin (b x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Move the limit inside the trig function because sine is continuous.

$\frac{0}{\sin (lim_{x \to 0} b x)}$

Step 1.1.3.1.2

Move the term $b$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\sin (b lim_{x \to 0} x)}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\sin (b \cdot 0)}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $b$ by $0$ .

$\frac{0}{\sin (0)}$

Step 1.1.3.3.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{\ln (\tan (45 + a x))}{\sin (b x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\tan (45 + a x))]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\tan (45 + a x))]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \tan (45 + a x)$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u_{1}$ as $\tan (45 + a x)$ .

$lim_{x \to 0} \frac{\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$lim_{x \to 0} \frac{\frac{1}{u_{1}} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.2.3

Replace all occurrences of $u_{1}$ with $\tan (45 + a x)$ .

$lim_{x \to 0} \frac{\frac{1}{\tan (45 + a x)} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.3

Rewrite $\tan (45 + a x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{1}{\frac{\sin (45 + a x)}{\cos (45 + a x)}} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (45 + a x)}{\cos (45 + a x)}$ .

$lim_{x \to 0} \frac{1 \frac{\cos (45 + a x)}{\sin (45 + a x)} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.5

Write $1$ as a fraction with denominator $1$ .

$lim_{x \to 0} \frac{\frac{1}{1} \cdot \frac{\cos (45 + a x)}{\sin (45 + a x)} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.6

Simplify.

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Step 1.3.6.1

Rewrite the expression.

$lim_{x \to 0} \frac{1 \frac{\cos (45 + a x)}{\sin (45 + a x)} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.6.2

Multiply $\frac{\cos (45 + a x)}{\sin (45 + a x)}$ by $1$ .

$lim_{x \to 0} \frac{\frac{\cos (45 + a x)}{\sin (45 + a x)} \frac{d}{d x} [\tan (45 + a x)]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 45 + a x$ .

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Step 1.3.7.1

To apply the Chain Rule, set $u_{2}$ as $45 + a x$ .

$lim_{x \to 0} \frac{\frac{\cos (45 + a x)}{\sin (45 + a x)} (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [45 + a x])}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.7.2

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

$lim_{x \to 0} \frac{\frac{\cos (45 + a x)}{\sin (45 + a x)} (\sec^{2} (u_{2}) \frac{d}{d x} [45 + a x])}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.7.3

Replace all occurrences of $u_{2}$ with $45 + a x$ .

$lim_{x \to 0} \frac{\frac{\cos (45 + a x)}{\sin (45 + a x)} (\sec^{2} (45 + a x) \frac{d}{d x} [45 + a x])}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.8

Combine $\sec^{2} (45 + a x)$ and $\frac{\cos (45 + a x)}{\sin (45 + a x)}$ .

$lim_{x \to 0} \frac{\frac{\sec^{2} (45 + a x) \cos (45 + a x)}{\sin (45 + a x)} \frac{d}{d x} [45 + a x]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.9

By the Sum Rule, the derivative of $45 + a x$ with respect to $x$ is $\frac{d}{d x} [45] + \frac{d}{d x} [a x]$ .

$lim_{x \to 0} \frac{\frac{\sec^{2} (45 + a x) \cos (45 + a x)}{\sin (45 + a x)} (\frac{d}{d x} [45] + \frac{d}{d x} [a x])}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.10

Since $45$ is constant with respect to $x$ , the derivative of $45$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{\frac{\sec^{2} (45 + a x) \cos (45 + a x)}{\sin (45 + a x)} (0 + \frac{d}{d x} [a x])}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.11

Add $0$ and $\frac{d}{d x} [a x]$ .

$lim_{x \to 0} \frac{\frac{\sec^{2} (45 + a x) \cos (45 + a x)}{\sin (45 + a x)} \frac{d}{d x} [a x]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.12

Since $a$ is constant with respect to $x$ , the derivative of $a x$ with respect to $x$ is $a \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{\sec^{2} (45 + a x) \cos (45 + a x)}{\sin (45 + a x)} (a \frac{d}{d x} [x])}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.13

Combine $a$ and $\frac{\sec^{2} (45 + a x) \cos (45 + a x)}{\sin (45 + a x)}$ .

$lim_{x \to 0} \frac{\frac{a (\sec^{2} (45 + a x) \cos (45 + a x))}{\sin (45 + a x)} \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.14

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

$lim_{x \to 0} \frac{\frac{a (\sec^{2} (45 + a x) \cos (45 + a x))}{\sin (45 + a x)} \cdot 1}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.15

Multiply $\frac{a (\sec^{2} (45 + a x) \cos (45 + a x))}{\sin (45 + a x)}$ by $1$ .

$lim_{x \to 0} \frac{\frac{a (\sec^{2} (45 + a x) \cos (45 + a x))}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16

Simplify.

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Step 1.3.16.1

Simplify the numerator.

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Step 1.3.16.1.1

Rewrite $\sec (45 + a x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{a ({(\frac{1}{\cos (45 + a x)})}^{2} \cos (45 + a x))}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.1.2

Apply the product rule to $\frac{1}{\cos (45 + a x)}$ .

$lim_{x \to 0} \frac{\frac{a (\frac{1^{2}}{\cos^{2} (45 + a x)} \cos (45 + a x))}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.1.3

Cancel the common factor of $\cos (45 + a x)$ .

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Step 1.3.16.1.3.1

Factor $\cos (45 + a x)$ out of $\cos^{2} (45 + a x)$ .

$lim_{x \to 0} \frac{\frac{a (\frac{1^{2}}{\cos (45 + a x) \cos (45 + a x)} \cos (45 + a x))}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.1.3.2

Cancel the common factor.

$lim_{x \to 0} \frac{\frac{a (\frac{1^{2}}{\cos (45 + a x) \cos (45 + a x)} \cos (45 + a x))}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.1.3.3

Rewrite the expression.

$lim_{x \to 0} \frac{\frac{a \frac{1^{2}}{\cos (45 + a x)}}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.1.4

One to any power is one.

$lim_{x \to 0} \frac{\frac{a \frac{1}{\cos (45 + a x)}}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.1.5

Combine $a$ and $\frac{1}{\cos (45 + a x)}$ .

$lim_{x \to 0} \frac{\frac{\frac{a}{\cos (45 + a x)}}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.2

Combine terms.

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Step 1.3.16.2.1

Rewrite $\frac{\frac{a}{\cos (45 + a x)}}{\sin (45 + a x)}$ as a product.

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x)} \cdot \frac{1}{\sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.16.2.2

Multiply $\frac{a}{\cos (45 + a x)}$ by $\frac{1}{\sin (45 + a x)}$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\frac{d}{d x} [\sin (b x)]}$

Step 1.3.17

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

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Step 1.3.17.1

To apply the Chain Rule, set $u$ as $b x$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [b x]}$

Step 1.3.17.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\cos (u) \frac{d}{d x} [b x]}$

Step 1.3.17.3

Replace all occurrences of $u$ with $b x$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\cos (b x) \frac{d}{d x} [b x]}$

Step 1.3.18

Since $b$ is constant with respect to $x$ , the derivative of $b x$ with respect to $x$ is $b \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\cos (b x) (b \frac{d}{d x} [x])}$

Step 1.3.19

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

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\cos (b x) (b \cdot 1)}$

Step 1.3.20

Multiply $b$ by $1$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{\cos (b x) b}$

Step 1.3.21

Reorder the factors of $\cos (b x) b$ .

$lim_{x \to 0} \frac{\frac{a}{\cos (45 + a x) \sin (45 + a x)}}{b \cos (b x)}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{a}{\cos (45 + a x) \sin (45 + a x)} \cdot \frac{1}{b \cos (b x)}$

Step 1.5

Multiply $\frac{a}{\cos (45 + a x) \sin (45 + a x)}$ by $\frac{1}{b \cos (b x)}$ .

$lim_{x \to 0} \frac{a}{\cos (45 + a x) \sin (45 + a x) (b \cos (b x))}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{a}{b}$ outside of the limit because it is constant with respect to $x$ .

$\frac{a}{b} lim_{x \to 0} \frac{1}{\cos (45 + a x) \sin (45 + a x) (\cos (b x))}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} \cos (45 + a x) \sin (45 + a x) (\cos (b x))}$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{1}{lim_{x \to 0} \cos (45 + a x) \sin (45 + a x) (\cos (b x))}$

Step 2.4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{1}{lim_{x \to 0} \cos (45 + a x) \cdot lim_{x \to 0} \sin (45 + a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.5

Move the limit inside the trig function because cosine is continuous.

$\frac{a}{b} \cdot \frac{1}{\cos (lim_{x \to 0} 45 + a x) \cdot lim_{x \to 0} \sin (45 + a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{1}{\cos (lim_{x \to 0} 45 + lim_{x \to 0} a x) \cdot lim_{x \to 0} \sin (45 + a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.7

Evaluate the limit of $45$ which is constant as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + lim_{x \to 0} a x) \cdot lim_{x \to 0} \sin (45 + a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.8

Move the term $a$ outside of the limit because it is constant with respect to $x$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (45 + a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.9

Move the limit inside the trig function because sine is continuous.

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 45 + a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.10

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 45 + lim_{x \to 0} a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.11

Evaluate the limit of $45$ which is constant as $x$ approaches $0$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot \sin (45 + lim_{x \to 0} a x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.12

Move the term $a$ outside of the limit because it is constant with respect to $x$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot \sin (45 + a lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (b x)}$

Step 2.13

Move the limit inside the trig function because cosine is continuous.

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot \sin (45 + a lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} b x)}$

Step 2.14

Move the term $b$ outside of the limit because it is constant with respect to $x$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a lim_{x \to 0} x) \cdot \sin (45 + a lim_{x \to 0} x) \cdot \cos (b lim_{x \to 0} x)}$

Step 3

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a \cdot 0) \cdot \sin (45 + a lim_{x \to 0} x) \cdot \cos (b lim_{x \to 0} x)}$

Step 3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a \cdot 0) \cdot \sin (45 + a \cdot 0) \cdot \cos (b lim_{x \to 0} x)}$

Step 3.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{a}{b} \cdot \frac{1}{\cos (45 + a \cdot 0) \cdot \sin (45 + a \cdot 0) \cdot \cos (b \cdot 0)}$

Step 4

Simplify the answer.

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Step 4.1

Separate fractions.

$\frac{a}{b} (\frac{1}{\sin (45 + a \cdot 0) \cdot \cos (b \cdot 0)} \cdot \frac{1}{\cos (45 + a \cdot 0)})$

Step 4.2

Convert from $\frac{1}{\cos (45 + a \cdot 0)}$ to $\sec (45 + a \cdot 0)$ .

$\frac{a}{b} (\frac{1}{\sin (45 + a \cdot 0) \cdot \cos (b \cdot 0)} \sec (45 + a \cdot 0))$

Step 4.3

Multiply $a$ by $0$ .

$\frac{a}{b} (\frac{1}{\sin (45 + 0) \cdot \cos (b \cdot 0)} \sec (45 + a \cdot 0))$

Step 4.4

Multiply $b$ by $0$ .

$\frac{a}{b} (\frac{1}{\sin (45 + 0) \cdot \cos (0)} \sec (45 + a \cdot 0))$

Step 4.5

Multiply $a$ by $0$ .

$\frac{a}{b} (\frac{1}{\sin (45 + 0) \cdot \cos (0)} \sec (45 + 0))$

Step 4.6

Separate fractions.

$\frac{a}{b} (\frac{1}{\cos (0)} \cdot \frac{1}{\sin (45 + 0)} \sec (45 + 0))$

Step 4.7

Convert from $\frac{1}{\sin (45 + 0)}$ to $\csc (45 + 0)$ .

$\frac{a}{b} (\frac{1}{\cos (0)} \csc (45 + 0) \sec (45 + 0))$

Step 4.8

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

$\frac{a}{b} (\sec (0) \csc (45 + 0) \sec (45 + 0))$

Step 4.9

The exact value of $\sec (0)$ is $1$ .

$\frac{a}{b} (1 \csc (45 + 0) \sec (45 + 0))$

Step 4.10

Multiply $\csc (45 + 0)$ by $1$ .

$\frac{a}{b} (\csc (45 + 0) \sec (45 + 0))$

Step 4.11

Add $45$ and $0$ .

$\frac{a}{b} (\csc (45) \sec (45 + 0))$

Step 4.12

The exact value of $\csc (45)$ is $\sqrt{2}$ .

$\frac{a}{b} (\sqrt{2} \sec (45 + 0))$

Step 4.13

Add $45$ and $0$ .

$\frac{a}{b} (\sqrt{2} \sec (45))$

Step 4.14

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

$\frac{a}{b} (\sqrt{2} \frac{2}{\sqrt{2}})$

Step 4.15

Cancel the common factor of $\sqrt{2}$ .

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Step 4.15.1

Cancel the common factor.

$\frac{a}{b} (\sqrt{2} \frac{2}{\sqrt{2}})$

Step 4.15.2

Rewrite the expression.

$\frac{a}{b} \cdot 2$

Step 4.16

Combine $\frac{a}{b}$ and $2$ .

$\frac{a \cdot 2}{b}$

Step 4.17

Move $2$ to the left of $a$ .

$\frac{2 a}{b}$