Trigonometry Examples

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

Step 1

Rearrange terms.

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

Step 2

Apply pythagorean identity.

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

Step 3

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

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

Step 4

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

${(\frac{\csc (\frac{π}{2} - x)}{\frac{1}{\cos (x)}})}^{2}$

Step 5

Rewrite $\csc (\frac{π}{2} - x)$ in terms of sines and cosines.

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

Step 6

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

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

Step 7

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Convert from $\frac{1}{\sin (\frac{π}{2} - x)}$ to $\csc (\frac{π}{2} - x)$ .

${(\csc (\frac{π}{2} - x) \cos (x))}^{2}$

Step 9

Apply the product rule to $\csc (\frac{π}{2} - x) \cos (x)$ .

$\csc^{2} (\frac{π}{2} - x) \cos^{2} (x)$