Trigonometry Examples

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

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{{(1 - \cos (x))}^{2}}{1 - \cos^{2} (x)}}$ as ${(\frac{{(1 - \cos (x))}^{2}}{1 - \cos^{2} (x)})}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{{(1 - \cos (x))}^{2}}{1 - \cos^{2} (x)})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify the denominator.

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Step 3.2.1.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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

Step 3.2.1.3

Reduce the expression by cancelling the common factors.

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Step 3.2.1.3.1

Cancel the common factor of ${(1 - \cos (x))}^{2}$ and $1 - \cos (x)$ .

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Step 3.2.1.3.1.1

Factor $1 - \cos (x)$ out of ${(1 - \cos (x))}^{2}$ .

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

Step 3.2.1.3.1.2

Cancel the common factors.

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Step 3.2.1.3.1.2.1

Factor $1 - \cos (x)$ out of $(1 + \cos (x)) (1 - \cos (x))$ .

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

Step 3.2.1.3.1.2.2

Cancel the common factor.

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

Step 3.2.1.3.1.2.3

Rewrite the expression.

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

Step 3.2.1.3.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the product rule to $\frac{1 - \cos (x)}{\sin (x)}$ .

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

Step 3.3.1.2

Multiply by $1$ .

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

Step 3.3.1.3

Separate fractions.

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

Step 3.3.1.4

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 3.3.1.5

Simplify the expression.

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Step 3.3.1.5.1

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

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

Step 3.3.1.5.2

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

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

Step 3.3.1.6

Expand $(1 - \cos (x)) (1 - \cos (x))$ using the FOIL Method.

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Step 3.3.1.6.1

Apply the distributive property.

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

Step 3.3.1.6.2

Apply the distributive property.

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

Step 3.3.1.6.3

Apply the distributive property.

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

Step 3.3.1.7

Simplify and combine like terms.

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Step 3.3.1.7.1

Simplify each term.

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Step 3.3.1.7.1.1

Multiply $1$ by $1$ .

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

Step 3.3.1.7.1.2

Multiply $- \cos (x)$ by $1$ .

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

Step 3.3.1.7.1.3

Multiply $- 1$ by $1$ .

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

Step 3.3.1.7.1.4

Multiply $- \cos (x) (- \cos (x))$ .

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Step 3.3.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.7.1.4.2

Multiply $\cos (x)$ by $1$ .

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

Step 3.3.1.7.1.4.3

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

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

Step 3.3.1.7.1.4.4

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

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

Step 3.3.1.7.1.4.5

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

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

Step 3.3.1.7.1.4.6

Add $1$ and $1$ .

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

Step 3.3.1.7.2

Subtract $\cos (x)$ from $- \cos (x)$ .

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

Step 3.3.1.8

Apply the distributive property.

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

Step 3.3.1.9

Simplify.

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Step 3.3.1.9.1

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

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

Step 3.3.1.9.2

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

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

Step 3.3.1.9.3

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 3.3.1.9.4

One to any power is one.

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

Step 3.3.1.9.5

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

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

Step 3.3.1.10

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

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

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $1 + \cos (x)$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $1 + \cos (x)$ .

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Step 4.2.1.1.1

Cancel the common factor.

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

Step 4.2.1.1.2

Rewrite the expression.

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

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Simplify each term.

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Step 4.2.2.1.1.1

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

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

Step 4.2.2.1.1.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 4.2.2.1.1.3

One to any power is one.

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

Step 4.2.2.1.1.4

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

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

Step 4.2.2.1.1.5

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 4.2.2.1.1.6

One to any power is one.

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

Step 4.2.2.1.1.7

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

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Step 4.2.2.1.1.7.1

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

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

Step 4.2.2.1.1.7.2

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

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

Step 4.2.2.1.1.8

Move the negative in front of the fraction.

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

Step 4.2.2.1.1.9

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

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

Step 4.2.2.1.1.10

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

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

Step 4.2.2.1.2

Expand $(\frac{1}{\sin^{2} (x)} - \frac{2 \cos (x)}{\sin^{2} (x)} + \frac{\cos^{2} (x)}{\sin^{2} (x)}) (1 + \cos (x))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.2.2.1.3

Simplify terms.

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Step 4.2.2.1.3.1

Simplify each term.

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Step 4.2.2.1.3.1.1

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

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

Step 4.2.2.1.3.1.2

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

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

Step 4.2.2.1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 4.2.2.1.3.1.4

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

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Step 4.2.2.1.3.1.4.1

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

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

Step 4.2.2.1.3.1.4.2

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

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

Step 4.2.2.1.3.1.4.3

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

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

Step 4.2.2.1.3.1.4.4

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

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

Step 4.2.2.1.3.1.4.5

Add $1$ and $1$ .

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

Step 4.2.2.1.3.1.5

Multiply $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ by $1$ .

Step 4.2.2.1.3.1.6

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

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Step 4.2.2.1.3.1.6.1

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

Step 4.2.2.1.3.1.6.2

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

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Step 4.2.2.1.3.1.6.2.1

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

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Step 4.2.2.1.3.1.6.2.1.1

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

Step 4.2.2.1.3.1.6.2.1.2

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

Step 4.2.2.1.3.1.6.2.2

Add $2$ and $1$ .

Step 4.2.2.1.3.2

Simplify terms.

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Step 4.2.2.1.3.2.1

Combine the numerators over the common denominator.

$1 - \cos (x) = \frac{1 + \cos (x) - 2 \cos (x) - 2 \cos^{2} (x) + \cos^{2} (x) + \cos^{3} (x)}{\sin^{2} (x)}$

Step 4.2.2.1.3.2.2

Subtract $2 \cos (x)$ from $\cos (x)$ .

$1 - \cos (x) = \frac{1 - \cos (x) - 2 \cos^{2} (x) + \cos^{2} (x) + \cos^{3} (x)}{\sin^{2} (x)}$

Step 4.2.2.1.3.2.3

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

$1 - \cos (x) = \frac{1 - \cos (x) - \cos^{2} (x) + \cos^{3} (x)}{\sin^{2} (x)}$

Step 4.2.2.1.4

Simplify the numerator.

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Step 4.2.2.1.4.1

Factor out the greatest common factor from each group.

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Step 4.2.2.1.4.1.1

Group the first two terms and the last two terms.

$1 - \cos (x) = \frac{(1 - \cos (x)) - \cos^{2} (x) + \cos^{3} (x)}{\sin^{2} (x)}$

Step 4.2.2.1.4.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.2.2.1.4.2

Factor the polynomial by factoring out the greatest common factor, $1 - \cos (x)$ .

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

Step 4.2.2.1.4.3

Rewrite $1$ as $1^{2}$ .

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

Step 4.2.2.1.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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

Step 4.2.2.1.4.5

Combine exponents.

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Step 4.2.2.1.4.5.1

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

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

Step 4.2.2.1.4.5.2

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

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

Step 4.2.2.1.4.5.3

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

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

Step 4.2.2.1.4.5.4

Add $1$ and $1$ .

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

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $\frac{{(1 - \cos (x))}^{2} (1 + \cos (x))}{\sin^{2} (x)}$ from both sides of the equation.

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

Step 4.3.2

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

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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

Step 4.3.2.3

Combine the numerators over the common denominator.

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

Step 4.3.2.4

Simplify each term.

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Step 4.3.2.4.1

Simplify the numerator.

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Step 4.3.2.4.1.1

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

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

Step 4.3.2.4.1.2

Expand $(1 - \cos (x)) (1 - \cos (x))$ using the FOIL Method.

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Step 4.3.2.4.1.2.1

Apply the distributive property.

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

Step 4.3.2.4.1.2.2

Apply the distributive property.

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

Step 4.3.2.4.1.2.3

Apply the distributive property.

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

Step 4.3.2.4.1.3

Simplify and combine like terms.

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Step 4.3.2.4.1.3.1

Simplify each term.

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Step 4.3.2.4.1.3.1.1

Multiply $1$ by $1$ .

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

Step 4.3.2.4.1.3.1.2

Multiply $- \cos (x)$ by $1$ .

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

Step 4.3.2.4.1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 4.3.2.4.1.3.1.4

Multiply $- \cos (x) (- \cos (x))$ .

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Step 4.3.2.4.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.4.1.3.1.4.2

Multiply $\cos (x)$ by $1$ .

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

Step 4.3.2.4.1.3.1.4.3

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

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

Step 4.3.2.4.1.3.1.4.4

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

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

Step 4.3.2.4.1.3.1.4.5

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

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

Step 4.3.2.4.1.3.1.4.6

Add $1$ and $1$ .

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

Step 4.3.2.4.1.3.2

Subtract $\cos (x)$ from $- \cos (x)$ .

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

Step 4.3.2.4.1.4

Apply the distributive property.

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

Step 4.3.2.4.1.5

Simplify.

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Step 4.3.2.4.1.5.1

Multiply $- 1$ by $1$ .

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

Step 4.3.2.4.1.5.2

Multiply $- 2$ by $- 1$ .

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

Step 4.3.2.4.1.6

Expand $(- 1 + 2 \cos (x) - \cos^{2} (x)) (1 + \cos (x))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.3.2.4.1.7

Simplify each term.

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Step 4.3.2.4.1.7.1

Multiply $- 1$ by $1$ .

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

Step 4.3.2.4.1.7.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 4.3.2.4.1.7.3

Multiply $2$ by $1$ .

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

Step 4.3.2.4.1.7.4

Multiply $2 \cos (x) \cos (x)$ .

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Step 4.3.2.4.1.7.4.1

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

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

Step 4.3.2.4.1.7.4.2

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

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

Step 4.3.2.4.1.7.4.3

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

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

Step 4.3.2.4.1.7.4.4

Add $1$ and $1$ .

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

Step 4.3.2.4.1.7.5

Multiply $- 1$ by $1$ .

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

Step 4.3.2.4.1.7.6

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

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Step 4.3.2.4.1.7.6.1

Move $\cos (x)$ .

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

Step 4.3.2.4.1.7.6.2

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

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Step 4.3.2.4.1.7.6.2.1

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

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

Step 4.3.2.4.1.7.6.2.2

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

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

Step 4.3.2.4.1.7.6.3

Add $1$ and $2$ .

$1 + \frac{- \cos (x) \sin^{2} (x) - 1 - \cos (x) + 2 \cos (x) + 2 \cos^{2} (x) - \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.8

Add $- \cos (x)$ and $2 \cos (x)$ .

$1 + \frac{- \cos (x) \sin^{2} (x) - 1 + \cos (x) + 2 \cos^{2} (x) - \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.9

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

$1 + \frac{- \cos (x) \sin^{2} (x) - 1 + \cos (x) + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.10

Move $\cos (x)$ .

$1 + \frac{- \cos (x) \sin^{2} (x) + \cos (x) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.11

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

$1 + \frac{\cos (x) - \cos (x) \sin^{2} (x) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.12

Multiply by $1$ .

$1 + \frac{\cos (x) \cdot 1 - \cos (x) \sin^{2} (x) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.13

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

$1 + \frac{\cos (x) \cdot 1 + \cos (x) (- \sin^{2} (x)) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.14

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

$1 + \frac{\cos (x) \cdot (1 - \sin^{2} (x)) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.15

Apply pythagorean identity.

$1 + \frac{\cos (x) \cdot \cos^{2} (x) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.16

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

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Step 4.3.2.4.1.16.1

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

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Step 4.3.2.4.1.16.1.1

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

$1 + \frac{\cos^{1} (x) \cdot \cos^{2} (x) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.16.1.2

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

$1 + \frac{{\cos (x)}^{1 + 2} - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.16.2

Add $1$ and $2$ .

$1 + \frac{\cos^{3} (x) - 1 + \cos^{2} (x) - \cos^{3} (x)}{\sin^{2} (x)} = 0$

Step 4.3.2.4.1.17

Combine the opposite terms in $\cos^{3} (x) - 1 + \cos^{2} (x) - \cos^{3} (x)$ .

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Step 4.3.2.4.1.17.1

Subtract $\cos^{3} (x)$ from $\cos^{3} (x)$ .

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

Step 4.3.2.4.1.17.2

Subtract $1$ from $0$ .

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

Step 4.3.2.4.1.18

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.3.2.4.1.19

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 4.3.2.4.1.20

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 4.3.2.4.1.21

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

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

Step 4.3.2.4.1.22

Apply pythagorean identity.

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

Step 4.3.2.4.2

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

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Step 4.3.2.4.2.1

Cancel the common factor.

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

Step 4.3.2.4.2.2

Divide $- 1$ by $1$ .

$1 - 1 = 0$

Step 4.3.2.5

Subtract $1$ from $1$ .

$0 = 0$

Step 4.3.3

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$