Calculus Examples

$\csc^{3} (x) \cdot \tan^{2} (x) = (\csc (x)) (1 + \tan^{2} (x))$

Step 1

Start on the right side.

$(\csc (x)) (1 + \tan^{2} (x))$

Step 2

Apply Pythagorean identity.

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Step 2.1

Rearrange terms.

$\csc (x) (\tan^{2} (x) + 1)$

Step 2.2

Apply pythagorean identity.

$\csc (x) \sec^{2} (x)$

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to $\csc (x)$ .

$\frac{1}{\sin (x)} \sec^{2} (x)$

Step 3.2

Apply the reciprocal identity to $\sec (x)$ .

$\frac{1}{\sin (x)} {(\frac{1}{\cos (x)})}^{2}$

Step 3.3

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

$\frac{1}{\sin (x)} \cdot \frac{1^{2}}{\cos^{2} (x)}$

Step 4

Simplify.

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Step 4.1

Combine.

$\frac{1 \cdot 1^{2}}{\sin (x) {\cos (x)}^{2}}$

Step 4.2

Multiply $1$ by $1^{2}$ by adding the exponents.

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Step 4.2.1

Multiply $1$ by $1^{2}$ .

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Step 4.2.1.1

Raise $1$ to the power of $1$ .

$\frac{1^{1} \cdot 1^{2}}{\sin (x) {\cos (x)}^{2}}$

Step 4.2.1.2

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

$\frac{1^{1 + 2}}{\sin (x) {\cos (x)}^{2}}$

Step 4.2.2

Add $1$ and $2$ .

$\frac{1^{3}}{\sin (x) {\cos (x)}^{2}}$

Step 4.3

One to any power is one.

$\frac{1}{\sin (x) \cos^{2} (x)}$

Step 5

Reorder terms.

$\frac{1}{\cos^{2} (x) \sin (x)}$

Step 6

Rewrite $\frac{1}{\cos^{2} (x) \sin (x)}$ as $\csc^{3} (x) \cdot \tan^{2} (x)$ .

$\csc^{3} (x) \cdot \tan^{2} (x)$

Step 7

Because the two sides have been shown to be equivalent, the equation is an identity.

$\csc^{3} (x) \cdot \tan^{2} (x) = (\csc (x)) (1 + \tan^{2} (x))$ is an identity