Algebra Examples

cot (- x) cos (- x) + sin (- x)

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

Step 1

Simplify each term.

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Step 1.1

Since

cot (- x)

$\cot (- x)$ is an odd function, rewrite

cot (- x)

$\cot (- x)$ as

- cot (x)

$- \cot (x)$ .

- cot (x) cos (- x) + sin (- x)

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

Step 1.2

Rewrite

cot (x)

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

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

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

Step 1.3

Since

cos (- x)

$\cos (- x)$ is an even function, rewrite

cos (- x)

$\cos (- x)$ as

cos (x)

$\cos (x)$ .

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

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

Step 1.4

Multiply

- \frac{cos (x)}{sin (x)} cos (x)

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

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Step 1.4.1

Combine

cos (x)

$\cos (x)$ and

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

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

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

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

Step 1.4.2

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 1.4.3

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 1.4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.4.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 1.5

Since

sin (- x)

$\sin (- x)$ is an odd function, rewrite

sin (- x)

$\sin (- x)$ as

- sin (x)

$- \sin (x)$ .

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

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

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

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

Step 2

Simplify each term.

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Step 2.1

Factor

cos (x)

$\cos (x)$ out of

{cos}^{2} (x)

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

- \frac{cos (x) cos (x)}{sin (x)} - sin (x)

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

Step 2.2

Separate fractions.

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

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

Step 2.3

Convert from

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

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

cot (x)

$\cot (x)$ .

- (\frac{cos (x)}{1} cot (x)) - sin (x)

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

Step 2.4

Divide

cos (x)

$\cos (x)$ by

1

$1$ .

- cos (x) cot (x) - sin (x)

$- \cos (x) \cot (x) - \sin (x)$

- cos (x) cot (x) - sin (x)

$- \cos (x) \cot (x) - \sin (x)$