Trigonometry Examples

$\frac{1}{\sin (x) (1 - \cos (x))} = \frac{(\cos (x) + 1)}{\sin^{3} (x)}$

Step 1

Start on the left side.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Reorder factors in $\sin (x) + \sin (x) (- \cos (x))$ .

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

Step 4

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

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

Step 5

Combine.

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

Step 6

Multiply $- \sin (x) - \sin (x) \cos (x)$ by $1$ .

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

Step 7

Simplify denominator.

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Step 7.1

Expand $(\sin (x) - \sin (x) \cos (x)) (- \sin (x) - \sin (x) \cos (x))$ using the FOIL Method.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Apply the distributive property.

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

Step 7.1.3

Apply the distributive property.

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

Step 7.2

Simplify and combine like terms.

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

Step 8

Apply Pythagorean identity.

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Step 8.1

Factor $- \sin^{2} (x)$ out of $- \sin^{2} (x)$ .

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

Step 8.2

Factor $- \sin^{2} (x)$ out of $\sin^{2} (x) \cos^{2} (x)$ .

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

Step 8.3

Factor $- \sin^{2} (x)$ out of $- \sin^{2} (x) (1) - \sin^{2} (x) (- 1 \cos^{2} (x))$ .

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

Step 8.4

Rewrite $- 1 \cos^{2} (x)$ as $- \cos^{2} (x)$ .

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

Step 8.5

Apply pythagorean identity.

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

Step 9

Simplify.

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Step 9.1

Multiply ${\sin (x)}^{2}$ by ${\sin (x)}^{2}$ by adding the exponents.

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Step 9.1.1

Move ${\sin (x)}^{2}$ .

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

Step 9.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9.1.3

Add $2$ and $2$ .

$\frac{- \sin (x) - \sin (x) \cos (x)}{- {\sin (x)}^{4}}$

Step 9.2

Factor $- \sin (x)$ out of $- \sin (x) - \sin (x) \cos (x)$ .

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Step 9.2.1

Factor $- \sin (x)$ out of $- \sin (x)$ .

$\frac{- \sin (x) (1) - \sin (x) \cos (x)}{- {\sin (x)}^{4}}$

Step 9.2.2

Factor $- \sin (x)$ out of $- \sin (x) \cos (x)$ .

$\frac{- \sin (x) (1) - \sin (x) (\cos (x))}{- {\sin (x)}^{4}}$

Step 9.2.3

Factor $- \sin (x)$ out of $- \sin (x) (1) - \sin (x) (\cos (x))$ .

$\frac{- \sin (x) (1 + \cos (x))}{- {\sin (x)}^{4}}$

Step 9.3

Dividing two negative values results in a positive value.

$\frac{\sin (x) (1 + \cos (x))}{{\sin (x)}^{4}}$

Step 9.4

Cancel the common factors.

$\frac{1 + \cos (x)}{\sin^{3} (x)}$

Step 10

Rewrite $\frac{1 + \cos (x)}{\sin^{3} (x)}$ as $\frac{(\cos (x) + 1)}{\sin^{3} (x)}$ .

$\frac{(\cos (x) + 1)}{\sin^{3} (x)}$

Step 11

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

$\frac{1}{\sin (x) (1 - \cos (x))} = \frac{\cos (x) + 1}{\sin^{3} (x)}$ is an identity