Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Multiply by $1$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

Reorder terms.

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

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

$\sec (x)$