Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.2

Multiply $\frac{1}{\sin (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$2 \sin (x)$