Trigonometry Examples

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

Step 1

Rewrite $\frac{(1 - \cos (θ)) (1 + \cos (θ))}{\cos^{2} (θ)}$ as ${(\frac{1}{\cos (θ)})}^{2} ((1 - \cos (θ)) (1 + \cos (θ)))$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $(1 - \cos (θ)) (1 + \cos (θ))$ .

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

Step 1.2

Factor the perfect power $\cos^{2} (θ)$ out of $\cos^{2} (θ)$ .

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

Step 1.3

Rearrange the fraction $\frac{1^{2} ((1 - \cos (θ)) (1 + \cos (θ)))}{\cos^{2} (θ) \cdot 1}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$\sec (θ) \sqrt{(1 - \cos (θ)) (1 + \cos (θ))}$