Precalculus Examples

$\frac{\sin (a)}{1 - \cos (a)} = \frac{1 + \cos (a)}{\sin (a)}$

Step 1

Start on the left side.

$\frac{\sin (a)}{1 - \cos (a)}$

Step 2

Multiply $\frac{\sin (a)}{1 - \cos (a)}$ by $\frac{1 + \cos (a)}{1 + \cos (a)}$ .

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

Step 3

Combine.

$\frac{\sin (a) (1 + \cos (a))}{(1 - \cos (a)) (1 + \cos (a))}$

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\sin (a)$ by $1$ .

$\frac{\sin (a) + \sin (a) \cos (a)}{(1 - \cos (a)) (1 + \cos (a))}$

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 - \cos (a)) (1 + \cos (a))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{\sin (a) + \sin (a) \cos (a)}{1 (1 + \cos (a)) - \cos (a) (1 + \cos (a))}$

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply pythagorean identity.

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

Step 7

Simplify.

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Step 7.1

Factor $\sin (a)$ out of $\sin (a) + \sin (a) \cos (a)$ .

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Step 7.1.1

Multiply by $1$ .

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

Step 7.1.2

Factor $\sin (a)$ out of $\sin (a) \cos (a)$ .

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

Step 7.1.3

Factor $\sin (a)$ out of $\sin (a) \cdot 1 + \sin (a) (\cos (a))$ .

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

Step 7.2

Cancel the common factors.

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

Step 8

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

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