Trigonometry Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Rewrite

\frac{{tan}^{2} (t)}{{sec}^{2} (t)}

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

{(\frac{tan (t)}{sec (t)})}^{2}

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

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

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

Step 1.2

Rewrite

sec (t)

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

{⎛ ⎝ \frac{tan (t)}{\frac{1}{cos (t)}} ⎞ ⎠}^{2} + \frac{{cot}^{2} (t)}{{csc}^{2} (t)}

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

Step 1.3

Rewrite

tan (t)

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

{⎛ ⎜ ⎝ \frac{\frac{sin (t)}{cos (t)}}{\frac{1}{cos (t)}} ⎞ ⎟ ⎠}^{2} + \frac{{cot}^{2} (t)}{{csc}^{2} (t)}

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

Step 1.4

Multiply by the reciprocal of the fraction to divide by

\frac{1}{cos (t)}

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

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

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

Step 1.5

Write

cos (t)

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

1

$1$ .

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

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

Step 1.6

Cancel the common factor of

cos (t)

$\cos (t)$ .

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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.

$1$