Precalculus Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\cot^{2} (x)$ from both sides of the equation.

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

Step 1.2

Add $\cos^{2} (x)$ to both sides of the equation.

$\cot^{2} (x) \cos^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

${(\frac{\cos (x)}{\sin (x)})}^{2} \cos^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 2.1.2

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

$\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 2.1.3

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

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Step 2.1.3.1

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

$\frac{\cos^{2} (x) \cos^{2} (x)}{\sin^{2} (x)} - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 2.1.3.2

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

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Step 2.1.3.2.1

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

$\frac{{\cos (x)}^{2 + 2}}{\sin^{2} (x)} - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 2.1.3.2.2

Add $2$ and $2$ .

$\frac{\cos^{4} (x)}{\sin^{2} (x)} - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 2.1.4

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

$\frac{\cos^{4} (x)}{\sin^{2} (x)} - {(\frac{\cos (x)}{\sin (x)})}^{2} + \cos^{2} (x) = 0$

Step 2.1.5

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

$\frac{\cos^{4} (x)}{\sin^{2} (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

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

$\frac{\cos^{2} (x) \cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 2.2.2

Multiply by $1$ .

$\frac{\cos^{2} (x) \cos^{2} (x)}{\sin^{2} (x) \cdot 1} - \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 2.2.3

Separate fractions.

$\frac{\cos^{2} (x)}{1} \cdot \frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 2.2.4

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

$\frac{\cos^{2} (x)}{1} \cot^{2} (x) - \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 2.2.5

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

$\cos^{2} (x) \cot^{2} (x) - \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 2.2.6

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

$\cos^{2} (x) \cot^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 3

Replace the $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$(1 - \sin^{2} (x)) \cot^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$1 \cot^{2} (x) - \sin^{2} (x) \cot^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 4.2

Multiply $\cot^{2} (x)$ by $1$ .

$\cot^{2} (x) - \sin^{2} (x) \cot^{2} (x) - \cot^{2} (x) + \cos^{2} (x) = 0$

Step 5

Combine the opposite terms in $\cot^{2} (x) - \sin^{2} (x) \cot^{2} (x) - \cot^{2} (x) + \cos^{2} (x)$ .

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Step 5.1

Subtract $\cot^{2} (x)$ from $\cot^{2} (x)$ .

$- \sin^{2} (x) \cot^{2} (x) + 0 + \cos^{2} (x) = 0$

Step 5.2

Add $- \sin^{2} (x) \cot^{2} (x)$ and $0$ .

$- \sin^{2} (x) \cot^{2} (x) + \cos^{2} (x) = 0$

Step 6

Simplify the left side.

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Step 6.1

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

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Step 6.1.1

Simplify each term.

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Step 6.1.1.1

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

$- \sin^{2} (x) {(\frac{\cos (x)}{\sin (x)})}^{2} + \cos^{2} (x) = 0$

Step 6.1.1.2

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

$- \sin^{2} (x) \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 6.1.1.3

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

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Step 6.1.1.3.1

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

$\sin^{2} (x) \cdot - 1 \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 6.1.1.3.2

Cancel the common factor.

$\sin^{2} (x) \cdot - 1 \frac{\cos^{2} (x)}{\sin^{2} (x)} + \cos^{2} (x) = 0$

Step 6.1.1.3.3

Rewrite the expression.

$- 1 \cos^{2} (x) + \cos^{2} (x) = 0$

Step 6.1.1.4

Rewrite $- 1 \cos^{2} (x)$ as $- \cos^{2} (x)$ .

$- \cos^{2} (x) + \cos^{2} (x) = 0$

Step 6.1.2

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

$0 = 0$

Step 7

Since $0 = 0$ , the equation will always be true.

Always true