Trigonometry Examples

$\frac{1}{\sec (x) - \tan (x)} = \sec (x) + \tan (x)$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Combine.

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

Step 6

Simplify numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 7

Simplify denominator.

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Step 7.1

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

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Apply the distributive property.

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

Step 7.1.3

Apply the distributive property.

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

Step 7.2

Simplify and combine like terms.

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

Step 8

Apply pythagorean identity.

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

Step 9

Simplify.

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Step 9.1

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

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Step 9.1.1

Multiply by $1$ .

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.2

Cancel the common factors.

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

Step 10

Now consider the right side of the equation.

$\sec (x) + \tan (x)$

Step 11

Convert to sines and cosines.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

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

$\frac{1}{\sec (x) - \tan (x)} = \sec (x) + \tan (x)$ is an identity