Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

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

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

Step 3

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

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

Step 4

Separate fractions.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Add $1$ and $1$ .

$\cos^{2} (\frac{π}{2} - x) \sec (x)$