Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.1.3

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

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

Step 1.1.4

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.1.5

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

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Step 1.1.5.1

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

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

Step 1.1.5.2

Multiply $\cos^{2} (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.1.5.2.1

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

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

Step 1.1.5.2.2

Add $2$ and $2$ .

$1 + \frac{\cos^{2} (x)}{\sin^{2} (x)} - \cos^{2} (x) - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Step 1.2

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

$1 - \cos^{2} (x) + \frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Step 2

Apply pythagorean identity.

$\sin^{2} (x) + \frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Step 3

Simplify each term.

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Step 3.1

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

$\sin^{2} (x) + \cot^{2} (x) - \frac{\cos^{4} (x)}{\sin^{2} (x)}$

Step 3.2

Factor $\cos^{2} (x)$ out of $\cos^{4} (x)$ .

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

Step 3.3

Multiply by $1$ .

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

Step 3.4

Separate fractions.

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

Step 3.5

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

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

Step 3.6

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

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