Trigonometry Examples

$\sec (θ) - \cos (θ)$

Step 1

Rewrite $\sec (θ)$ in terms of sines and cosines.

$\frac{1}{\cos (θ)} - \cos (θ)$

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Rewrite $\cos (θ) \cos (θ)$ as $\cos^{2} (θ)$ .

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

Step 4.3

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 $b = \cos (θ)$ .

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