Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 1.4

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

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

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

Step 1.6.4

Add $1$ and $1$ .

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

Step 1.7

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

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

Step 1.8

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.9

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.10

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

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Step 1.10.1

Cancel the common factor.

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

Step 1.10.2

Rewrite the expression.

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

Separate fractions.

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

Step 2.3

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

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

Step 2.4

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

$\sin (x) \tan (x) + \cos (x)$