Trigonometry Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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 $b = \sin (θ)$ .

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

Step 2

Multiply $\frac{\sin (θ)}{\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}$ by $\frac{\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}{\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sin (θ)}{\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}$ by $\frac{\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}{\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}$ .

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

Step 3.2

Raise $\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}$ to the power of $1$ .

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

Step 3.3

Raise $\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}$ to the power of $1$ .

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

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Rewrite ${\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}}^{2}$ as $(1 + \sin (θ)) (1 - \sin (θ))$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + \sin (θ)) (1 - \sin (θ))}$ as ${((1 + \sin (θ)) (1 - \sin (θ)))}^{\frac{1}{2}}$ .

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

Step 3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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