Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify.

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Step 2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (θ)$ and $b = \sin (θ)$ .

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

Step 2.2

Cancel the common factor of $\cos (θ) + \sin (θ)$ .

$\cos (θ) - \sin (θ)$

Step 3

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

$\frac{\cos^{2} (θ) - \sin^{2} (θ)}{\cos (θ) + \sin (θ)} = \cos (θ) - \sin (θ)$ is an identity