Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

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

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 5

Simplify denominator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply pythagorean identity.

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

Step 7

Convert to sines and cosines.

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Step 7.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 7.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

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

Step 8.1.2

Combine the numerators over the common denominator.

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

Step 8.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3

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

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

Step 9

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

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

Step 10

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

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