Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

One to any power is one.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Cancel the common factor of $\sin (x)$ and $\sin^{2} (x)$ .

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Step 6.1

Factor $\sin (x)$ out of $\sin (x) \cos^{2} (x)$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $\sin (x)$ out of $\cos^{2} (x) \sin^{2} (x)$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $\cos^{2} (x)$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{1}{\sin (x)}$

Step 8

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x)$