Trigonometry Examples

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

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Multiply by $1$ .

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

Step 1.2.3

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

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

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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