Trigonometry Examples

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

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

Step 1

Rewrite

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

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

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

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

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Step 1.1

Factor the perfect power

1^{2}

$1^{2}$ out of

(1 - cos (θ)) (1 + cos (θ))

$(1 - \cos (θ)) (1 + \cos (θ))$ .

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

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

Step 1.2

Factor the perfect power

{cos}^{2} (θ)

$\cos^{2} (θ)$ out of

{cos}^{2} (θ)

$\cos^{2} (θ)$ .

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

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

Step 1.3

Rearrange the fraction

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

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

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

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

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

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

Step 2

Pull terms out from under the radical.

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

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

Step 3

Convert from

\frac{1}{cos (θ)}

$\frac{1}{\cos (θ)}$ to

sec (θ)

$\sec (θ)$ .

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

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