Trigonometry Examples

$\frac{1 - \sin^{2} (x)}{1 - \sin (x)} = \frac{\csc (x) + 1}{\csc (x)}$

Step 1

Start on the left side.

$\frac{1 - \sin^{2} (x)}{1 - \sin (x)}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the numerator.

Tap for more steps...

Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - {\sin (x)}^{2}}{1 - \sin (x)}$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

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

Step 2.2

Cancel the common factor of $1 - \sin (x)$ .

$1 + \sin (x)$

Step 3

Rewrite $1 + \sin (x)$ as $\frac{\csc (x) + 1}{\csc (x)}$ .

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

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1 - \sin^{2} (x)}{1 - \sin (x)} = \frac{\csc (x) + 1}{\csc (x)}$ is an identity