Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Combine.

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

Step 6

Simplify numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 7

Simplify denominator.

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Step 7.1

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

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Apply the distributive property.

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

Step 7.1.3

Apply the distributive property.

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

Step 7.2

Simplify and combine like terms.

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

Step 8

Apply pythagorean identity.

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

Step 9

Simplify.

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Step 9.1

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

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Step 9.1.1

Multiply by $1$ .

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.2

Cancel the common factors.

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

Step 10

Now consider the right side of the equation.

$\csc (x) + \cot (x)$

Step 11

Convert to sines and cosines.

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Step 11.1

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

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

Step 11.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

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

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