Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

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

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

Step 2.2.2

Multiply by $1$ .

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

Step 2.2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Cancel the common factor of $1 + \sec (x)$ and $\sec (x) + 1$ .

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Step 5.1

Reorder terms.

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

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

$\csc (x)$