Trigonometry Examples

$\frac{\tan (x) - \sin (x)}{\sin (x) \tan (x)}$

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)} - \sin (x)$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Factor $\sin (x)$ out of $\sin (x) \frac{1}{\cos (x)} + \sin (x) \cdot - 1$ .

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

Step 2

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

$\frac{\sin (x) (\frac{1}{\cos (x)} - 1)}{\sin (x) \frac{\sin (x)}{\cos (x)}}$

Step 3

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

$\frac{\sin (x) (\frac{1}{\cos (x)} - 1)}{\frac{\sin (x) \sin (x)}{\cos (x)}}$

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Simplify terms.

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Step 6.1

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

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Step 6.1.1

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

$\sin (x) (\frac{1}{\cos (x)} - 1) \frac{\cos (x)}{\sin (x) \sin (x)}$

Step 6.1.2

Cancel the common factor.

$\sin (x) (\frac{1}{\cos (x)} - 1) \frac{\cos (x)}{\sin (x) \sin (x)}$

Step 6.1.3

Rewrite the expression.

$(\frac{1}{\cos (x)} - 1) \frac{\cos (x)}{\sin (x)}$

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

$\frac{1}{\sin (x)} - 1 \frac{\cos (x)}{\sin (x)}$

Step 6.4

Rewrite $- 1 \frac{\cos (x)}{\sin (x)}$ as $- \frac{\cos (x)}{\sin (x)}$ .

$\frac{1}{\sin (x)} - \frac{\cos (x)}{\sin (x)}$

Step 7

Simplify each term.

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Step 7.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 7.2

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

$\csc (x) - \cot (x)$