Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

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 $b = \cos (x)$ .

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

Step 3.2

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

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

Step 4

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

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