Precalculus Examples

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

Step 1

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 2

Apply the distributive property.

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

Step 3

Combine $\cos (\frac{π}{2} - x)$ and $\frac{1}{\sin (x)}$ .

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

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

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

Step 6

Simplify terms.

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Step 6.1

Combine $- \cos (\frac{π}{2} - x) \sin (x)$ and $\frac{\sin (x)}{\sin (x)}$ .

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

Step 6.2

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 7.1.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 7.1.3

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

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

Step 7.1.4

Add $1$ and $1$ .

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

Step 7.2

Multiply by $1$ .

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

Step 7.3

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

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

Step 7.4

Factor $\cos (\frac{π}{2} - x)$ out of $\cos (\frac{π}{2} - x) \cdot 1 + \cos (\frac{π}{2} - x) (- \sin^{2} (x))$ .

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

Step 7.5

Apply pythagorean identity.

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

Step 8

Separate fractions.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 12

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

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