Precalculus Examples

$\frac{\cos (θ)}{1 - \sin (θ)} = \sec (θ) + \tan (θ)$

Step 1

Start on the left side.

$\frac{\cos (θ)}{1 - \sin (θ)}$

Step 2

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

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\cos (θ)$ by $1$ .

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

Step 5

Simplify denominator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply pythagorean identity.

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

Step 7

Simplify.

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Step 7.1

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

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Step 7.1.1

Multiply by $1$ .

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.2

Cancel the common factors.

$\frac{1 + \sin (θ)}{\cos (θ)}$

Step 8

Now consider the right side of the equation.

$\sec (θ) + \tan (θ)$

Step 9

Convert to sines and cosines.

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Step 9.1

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

$\frac{1}{\cos (θ)} + \tan (θ)$

Step 9.2

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

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

Step 10

Combine the numerators over the common denominator.

$\frac{1 + \sin (θ)}{\cos (θ)}$

Step 11

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

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