Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\frac{\tan^{2} (t)}{\sec^{2} (t)}$ as ${(\frac{\tan (t)}{\sec (t)})}^{2}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Cancel the common factor of $\cos (t)$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Rewrite the expression.

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

Step 1.7

Rewrite $\frac{\cot^{2} (t)}{\csc^{2} (t)}$ as ${(\frac{\cot (t)}{\csc (t)})}^{2}$ .

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Write $\sin (t)$ as a fraction with denominator $1$ .

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

Step 1.12

Cancel the common factor of $\sin (t)$ .

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Step 1.12.1

Cancel the common factor.

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

Step 1.12.2

Rewrite the expression.

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

Step 2

Apply pythagorean identity.

$1$