Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite

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

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

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

Step 1.2

Rewrite

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

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

Step 1.3

Rewrite

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

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

Step 1.4

Multiply by the reciprocal of the fraction to divide by

$\frac{1}{\cos (x)}$ .

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

Step 1.5

Write

$\cos (x)$ as a fraction with denominator

$1$ .

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

Step 1.6

Cancel the common factor of

$\cos (x)$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Rewrite the expression.

$\sin^{2} (x) + \frac{\cot^{2} (x)}{\csc^{2} (x)}$

Step 1.7

Rewrite

$\frac{\cot^{2} (x)}{\csc^{2} (x)}$ as

${(\frac{\cot (x)}{\csc (x)})}^{2}$ .

$\sin^{2} (x) + {(\frac{\cot (x)}{\csc (x)})}^{2}$

Step 1.8

Rewrite

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

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

Step 1.9

Rewrite

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

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

Step 1.10

Multiply by the reciprocal of the fraction to divide by

$\frac{1}{\sin (x)}$ .

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

Step 1.11

Write

$\sin (x)$ as a fraction with denominator

$1$ .

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

Step 1.12

Cancel the common factor of

$\sin (x)$ .

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Step 1.12.1

Cancel the common factor.

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

Step 1.12.2

Rewrite the expression.

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

Step 2

Apply pythagorean identity.

$1$