Calculus Examples

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 8

Multiply $\cos (x)$ by $1$ .

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

Step 9

Divide $\tan^{2} (x)$ by $1$ .

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

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

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Step 13.1

Factor $\cos (x)$ out of $\sin (x) \cos (x)$ .

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

Step 13.2

Cancel the common factor.

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

Step 13.3

Rewrite the expression.

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

Step 14

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

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Step 14.1

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

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Step 14.1.1

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

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

Step 14.1.2

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

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

Step 14.2

Add $2$ and $1$ .

$\sin^{3} (x)$