Trigonometry Examples

$\frac{\tan (t)}{\cot (t)} \cdot \frac{\cos (t)}{\sec (t)}$

Step 1

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

$\frac{\frac{\sin (t)}{\cos (t)}}{\cot (t)} \cdot \frac{\cos (t)}{\sec (t)}$

Step 2

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

$\frac{\frac{\sin (t)}{\cos (t)}}{\frac{\cos (t)}{\sin (t)}} \cdot \frac{\cos (t)}{\sec (t)}$

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{\sin (t)}{\cos (t)} \cdot \frac{\sin (t)}{\cos (t)}$ .

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Step 5.1

Multiply $\frac{\sin (t)}{\cos (t)}$ by $\frac{\sin (t)}{\cos (t)}$ .

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

Step 5.2

Raise $\sin (t)$ to the power of $1$ .

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

Step 5.3

Raise $\sin (t)$ to the power of $1$ .

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

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

Raise $\cos (t)$ to the power of $1$ .

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

Step 5.7

Raise $\cos (t)$ to the power of $1$ .

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

Step 5.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.9

Add $1$ and $1$ .

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

Step 6

Combine.

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

Step 7

Cancel the common factor of $\cos (t)$ and $\cos^{2} (t)$ .

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Step 7.1

Factor $\cos (t)$ out of $\sin^{2} (t) \cos (t)$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $\cos (t)$ out of $\cos^{2} (t) \frac{1}{\cos (t)}$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

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

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Reduce the expression $\frac{\cos (t)}{\cos (t)}$ by cancelling the common factors.

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Step 9.1.1

Cancel the common factor.

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

Step 9.1.2

Rewrite the expression.

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

Step 9.2

Rewrite the expression.

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

Step 10

Divide $\sin^{2} (t)$ by $1$ .

$\sin^{2} (t)$