Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Rewrite $1$ as $1^{2}$ .

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

Step 1.7

Rewrite $\frac{1^{2}}{\csc^{2} (x)}$ as ${(\frac{1}{\csc (x)})}^{2}$ .

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 2

Rearrange terms.

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

Step 3

Apply pythagorean identity.

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