Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply by $1$ .

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

Step 2.2.3

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

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

Step 3

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

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Combine.

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

Step 7

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 7.1

Factor $\sin (x)$ out of $\sin^{2} (x) \cdot 1$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

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

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

Step 9

Separate fractions.

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

Step 10

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

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

Step 11

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{1}{\sec (x) + 1} \tan (x)$

Step 12

Combine $\frac{1}{\sec (x) + 1}$ and $\tan (x)$ .

$\frac{\tan (x)}{\sec (x) + 1}$