Trigonometry Examples

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

Step 1

Rearrange terms.

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

Step 2

Apply pythagorean identity.

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

Step 3

Apply pythagorean identity.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

$\tan^{2} (x)$