Trigonometry Examples

$\frac{\cot (x) \cos (x)}{\sin (x)} \cdot \frac{\tan (x)}{\frac{\sin (x)}{\cos (x) \tan (x)}}$

Step 1

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

$\frac{\frac{\cos (x)}{\sin (x)} \cos (x)}{\sin (x)} \cdot \frac{\tan (x)}{\frac{\sin (x)}{\cos (x) \tan (x)}}$

Step 2

Combine $\frac{\cos (x)}{\sin (x)}$ and $\cos (x)$ .

$\frac{\frac{\cos (x) \cos (x)}{\sin (x)}}{\sin (x)} \cdot \frac{\tan (x)}{\frac{\sin (x)}{\cos (x) \tan (x)}}$

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

Combine $\cos (x)$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

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

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

Step 7

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

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

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Reduce the expression $\frac{\sin (x)}{\sin (x)}$ by cancelling the common factors.

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Step 8.1.1

Cancel the common factor.

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

Step 8.1.2

Rewrite the expression.

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

Step 8.2

Rewrite the expression.

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

Step 9

Combine fractions.

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Step 9.1

Combine.

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

Step 9.2

Multiply $\frac{\cos^{2} (x)}{\sin (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 9.3

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

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

Step 10

Simplify the numerator.

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Step 10.1

Reduce the expression $\frac{\cos^{2} (x) \sin (x)}{\sin (x) \cos (x)}$ by cancelling the common factors.

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Step 10.1.1

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

$\frac{\frac{\cos (x) (\cos (x) \sin (x))}{\sin (x) \cos (x)}}{\sin (x)}$

Step 10.1.2

Factor $\cos (x)$ out of $\sin (x) \cos (x)$ .

$\frac{\frac{\cos (x) (\cos (x) \sin (x))}{\cos (x) \sin (x)}}{\sin (x)}$

Step 10.1.3

Cancel the common factor.

$\frac{\frac{\cos (x) (\cos (x) \sin (x))}{\cos (x) \sin (x)}}{\sin (x)}$

Step 10.1.4

Rewrite the expression.

$\frac{\frac{\cos (x) \sin (x)}{\sin (x)}}{\sin (x)}$

Step 10.2

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

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Step 10.2.1

Cancel the common factor.

$\frac{\frac{\cos (x) \sin (x)}{\sin (x)}}{\sin (x)}$

Step 10.2.2

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

$\frac{\cos (x)}{\sin (x)}$

Step 11

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x)$