Precalculus Examples

$(- \cos (x) + \sec (x)) \cot (x)$

Step 1

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

$(- \cos (x) + \frac{1}{\cos (x)}) \cot (x)$

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $\cos (x)$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Reorder $- \cos^{2} (x)$ and $1$ .

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

Step 6

Apply pythagorean identity.

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

Step 7

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 7.1

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

$\frac{\sin (x) \sin (x)}{\sin (x)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Multiply by $1$ .

$\frac{\sin (x) \sin (x)}{\sin (x) \cdot 1}$

Step 7.2.2

Cancel the common factor.

$\frac{\sin (x) \sin (x)}{\sin (x) \cdot 1}$

Step 7.2.3

Rewrite the expression.

$\frac{\sin (x)}{1}$

Step 7.2.4

Divide $\sin (x)$ by $1$ .

$\sin (x)$