Trigonometry Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

Multiply by $1$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.2

Separate fractions.

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

Step 1.3

Rewrite $\frac{\sin (x + 1)}{\cos (x)}$ as a product.

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

Step 1.4

Write $\sin (x + 1)$ as a fraction with denominator $1$ .

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Divide $\sin (x + 1)$ by $1$ .

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

Step 1.5.2

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

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

Step 1.6

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

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

Step 1.7

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

Tap for more steps...

Step 1.7.1

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

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

Step 1.7.2

Combine $\frac{\sin (x + 1)}{1 + \csc (x)}$ and $\sec (x)$ .

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

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = π n$ , for any integer $n$

Step 3