Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Simplify the numerator.

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Step 2.1

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.2

Combine $- x$ and $\frac{2}{2}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $x$ out of $x - x \cdot 2$ .

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Step 2.4.1.1

Raise $x$ to the power of $1$ .

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

Step 2.4.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.4.1.3

Factor $x$ out of $- x \cdot 2$ .

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

Step 2.4.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 2)$ .

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

Step 2.4.2

Multiply $- 1$ by $2$ .

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

Step 2.4.3

Subtract $2$ from $1$ .

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

Step 2.5

Move $- 1$ to the left of $x$ .

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

Step 2.6

Move the negative in front of the fraction.

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

Step 2.7

Since $\cos (- \frac{x}{2})$ is an even function, rewrite $\cos (- \frac{x}{2})$ as $\cos (\frac{x}{2})$ .

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

Step 3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\cos^{2} (\frac{x}{2})}{\cos (x)}$

Step 4

Factor $\cos (\frac{x}{2})$ out of $\cos^{2} (\frac{x}{2})$ .

$\frac{\cos (\frac{x}{2}) \cos (\frac{x}{2})}{\cos (x)}$

Step 5

Separate fractions.

$\frac{\cos (\frac{x}{2})}{1} \cdot \frac{\cos (\frac{x}{2})}{\cos (x)}$

Step 6

Rewrite $\frac{\cos (\frac{x}{2})}{\cos (x)}$ as a product.

$\frac{\cos (\frac{x}{2})}{1} (\cos (\frac{x}{2}) \frac{1}{\cos (x)})$

Step 7

Write $\cos (\frac{x}{2})$ as a fraction with denominator $1$ .

$\frac{\cos (\frac{x}{2})}{1} (\frac{\cos (\frac{x}{2})}{1} \cdot \frac{1}{\cos (x)})$

Step 8

Simplify.

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Step 8.1

Divide $\cos (\frac{x}{2})$ by $1$ .

$\frac{\cos (\frac{x}{2})}{1} (\cos (\frac{x}{2}) \frac{1}{\cos (x)})$

Step 8.2

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

$\frac{\cos (\frac{x}{2})}{1} (\cos (\frac{x}{2}) \sec (x))$

Step 9

Divide $\cos (\frac{x}{2})$ by $1$ .

$\cos (\frac{x}{2}) (\cos (\frac{x}{2}) \sec (x))$

Step 10

Multiply $\cos (\frac{x}{2}) (\cos (\frac{x}{2}) \sec (x))$ .

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Step 10.1

Raise $\cos (\frac{x}{2})$ to the power of $1$ .

$\cos^{1} (\frac{x}{2}) \cos (\frac{x}{2}) \sec (x)$

Step 10.2

Raise $\cos (\frac{x}{2})$ to the power of $1$ .

$\cos^{1} (\frac{x}{2}) \cos^{1} (\frac{x}{2}) \sec (x)$

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

$\cos^{2} (\frac{x}{2}) \sec (x)$