Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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