Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

To write $- \cos (x)$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Combine the numerators over the common denominator.

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

Step 3.1.4

Simplify the numerator.

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Step 3.1.4.1

Multiply $- \cos (x) \cos (x)$ .

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Step 3.1.4.1.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.1.4.1.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.1.4.1.3

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

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

Step 3.1.4.1.4

Add $1$ and $1$ .

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

Step 3.1.4.2

Rewrite $1 - {\cos (x)}^{2}$ in a factored form.

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Step 3.1.4.2.1

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

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

Step 3.1.4.2.2

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 $b = \cos (x)$ .

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 4

Apply the distributive property.

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

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $1 + \cos (x)$ by $1$ .

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

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

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

Step 5.1.4

Multiply $\cos (x) (- \cos (x))$ .

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Step 5.1.4.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 5.1.4.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 5.1.4.3

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

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

Step 5.1.4.4

Add $1$ and $1$ .

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

Step 5.2

Subtract $\cos (x)$ from $\cos (x)$ .

$\frac{\cos (x)}{1 + 0 - {\cos (x)}^{2}}$

Step 5.3

Add $1$ and $0$ .

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

Step 6

Apply pythagorean identity.

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

Step 7

Rewrite $\frac{\cos (x)}{\sin^{2} (x)}$ as $\cot (x) \csc (x)$ .

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

Step 8

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

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