Precalculus Examples

$\sqrt{\frac{\tan^{2} (x) + 1}{\cot^{2} (x) + 1}}$

Step 1

Apply pythagorean identity.

$\sqrt{\frac{\sec^{2} (x)}{\cot^{2} (x) + 1}}$

Step 2

Rearrange terms.

$\sqrt{\frac{\sec^{2} (x)}{1 + \cot^{2} (x)}}$

Step 3

Apply pythagorean identity.

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

Step 4

Rewrite $\frac{\sec^{2} (x)}{\csc^{2} (x)}$ as ${(\frac{\sec (x)}{\csc (x)})}^{2}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sqrt{\tan^{2} (x)}$

Step 9

Pull terms out from under the radical, assuming positive real numbers.

$\tan (x)$