Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

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

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

Step 2.2.2

Multiply by $1$ .

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

Step 2.2.3

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

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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