Trigonometry Examples

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

Step 1

Subtract $\frac{\sec (x) - \sin (x)}{\tan (x) - 1}$ from both sides of the equation.

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

Step 2

Simplify $\frac{\sin^{3} (x) + \cos^{3} (x)}{1 - 2 \cos^{2} (x)} - \frac{\sec (x) - \sin (x)}{\tan (x) - 1}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the numerator.

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Step 2.1.1.1

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = \sin (x)$ and $b = \cos (x)$ .

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

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Rearrange terms.

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

Step 2.1.1.2.2

Apply pythagorean identity.

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Step 2.1.4.1

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

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

Step 2.1.4.2

Combine.

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

Step 2.1.5

Apply the distributive property.

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

Step 2.1.6

Simplify by cancelling.

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Step 2.1.6.1

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

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Step 2.1.6.1.1

Cancel the common factor.

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

Step 2.1.6.1.2

Rewrite the expression.

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

Step 2.1.6.2

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

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Step 2.1.6.2.1

Cancel the common factor.

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

Step 2.1.6.2.2

Rewrite the expression.

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

Step 2.1.7

Rewrite using the commutative property of multiplication.

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

Step 2.1.8

Simplify the denominator.

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Step 2.1.8.1

Move $- 1$ to the left of $\cos (x)$ .

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

Step 2.1.8.2

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

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

Step 2.2

To write $\frac{(\sin (x) + \cos (x)) (- \sin (x) \cos (x) + 1)}{1 - 2 \cos^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x) - \cos (x)}{\sin (x) - \cos (x)}$ .

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

Step 2.3

To write $- \frac{1 - \cos (x) \sin (x)}{\sin (x) - \cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - 2 \cos^{2} (x)}{1 - 2 \cos^{2} (x)}$ .

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

Step 2.4

Write each expression with a common denominator of $(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Reorder the factors of $(\sin (x) - \cos (x)) (1 - 2 \cos^{2} (x))$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

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

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Step 2.6.1.1

Apply the distributive property.

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

Step 2.6.1.2

Apply the distributive property.

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

Step 2.6.1.3

Apply the distributive property.

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

Step 2.6.2

Simplify each term.

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Step 2.6.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.6.2.2

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

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Step 2.6.2.2.1

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

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

Step 2.6.2.2.2

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

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

Step 2.6.2.2.3

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

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

Step 2.6.2.2.4

Add $1$ and $1$ .

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

Step 2.6.2.3

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

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

Step 2.6.2.4

Rewrite using the commutative property of multiplication.

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

Step 2.6.2.5

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

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Step 2.6.2.5.1

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

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

Step 2.6.2.5.2

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

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

Step 2.6.2.5.3

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

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

Step 2.6.2.5.4

Add $1$ and $1$ .

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

Step 2.6.2.6

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

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

Step 2.6.3

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

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

Step 2.6.4

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

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Step 2.6.4.1

Reorder the factors in the terms $\sin (x) (- \cos (x))$ and $\cos (x) \sin (x)$ .

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

Step 2.6.4.2

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

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

Step 2.6.4.3

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

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

Step 2.6.5

Simplify each term.

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Step 2.6.5.1

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

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Step 2.6.5.1.1

Move $\sin (x)$ .

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

Step 2.6.5.1.2

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

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Step 2.6.5.1.2.1

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

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

Step 2.6.5.1.2.2

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

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

Step 2.6.5.1.3

Add $1$ and $2$ .

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

Step 2.6.5.2

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

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Step 2.6.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.5.2.2

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

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

Step 2.6.5.2.3

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

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

Step 2.6.5.2.4

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

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

Step 2.6.5.2.5

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

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

Step 2.6.5.2.6

Add $1$ and $1$ .

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

Step 2.6.5.3

Multiply $\sin (x) \sin (x)$ .

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Step 2.6.5.3.1

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

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

Step 2.6.5.3.2

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

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

Step 2.6.5.3.3

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

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

Step 2.6.5.3.4

Add $1$ and $1$ .

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

Step 2.6.5.4

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

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Step 2.6.5.4.1

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

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

Step 2.6.5.4.2

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

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

Step 2.6.5.4.3

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

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

Step 2.6.5.4.4

Add $1$ and $1$ .

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

Step 2.6.5.5

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

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Step 2.6.5.5.1

Move $\cos (x)$ .

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

Step 2.6.5.5.2

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

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Step 2.6.5.5.2.1

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

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

Step 2.6.5.5.2.2

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

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

Step 2.6.5.5.3

Add $1$ and $2$ .

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

Step 2.6.5.6

Multiply $- \cos^{3} (x) \sin (x) \cdot - 1$ .

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Step 2.6.5.6.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.5.6.2

Multiply $\cos^{3} (x)$ by $1$ .

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

Step 2.6.5.7

Rewrite using the commutative property of multiplication.

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

Step 2.6.5.8

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

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Step 2.6.5.8.1

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

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

Step 2.6.5.8.2

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

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

Step 2.6.5.8.3

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

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

Step 2.6.5.8.4

Add $1$ and $1$ .

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

Step 2.6.6

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

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Step 2.6.6.1

Reorder the factors in the terms $\sin^{2} (x) \cos^{2} (x)$ and $- \cos^{2} (x) \sin^{2} (x)$ .

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

Step 2.6.6.2

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

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

Step 2.6.6.3

Add $- \sin^{3} (x) \cos (x)$ and $0$ .

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

Step 2.6.7

Apply the distributive property.

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

Step 2.6.8

Multiply $- 1$ by $1$ .

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

Step 2.6.9

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

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Step 2.6.9.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.9.2

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

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

Step 2.6.10

Expand $(- 1 + \cos (x) \sin (x)) (1 - 2 \cos^{2} (x))$ using the FOIL Method.

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Step 2.6.10.1

Apply the distributive property.

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

Step 2.6.10.2

Apply the distributive property.

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

Step 2.6.10.3

Apply the distributive property.

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

Step 2.6.11

Simplify each term.

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Step 2.6.11.1

Multiply $- 1$ by $1$ .

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

Step 2.6.11.2

Multiply $- 2$ by $- 1$ .

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

Step 2.6.11.3

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

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

Step 2.6.11.4

Rewrite using the commutative property of multiplication.

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

Step 2.6.11.5

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

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Step 2.6.11.5.1

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

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

Step 2.6.11.5.2

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

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Step 2.6.11.5.2.1

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

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

Step 2.6.11.5.2.2

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

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

Step 2.6.11.5.3

Add $2$ and $1$ .

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

Step 2.6.12

Subtract $2 \cos^{3} (x) \sin (x)$ from $\cos^{3} (x) \sin (x)$ .

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

Step 2.6.13

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

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

Step 2.6.14

Rewrite $- \sin^{3} (x) \cos (x) + \sin^{2} (x) - \cos^{3} (x) \sin (x) - 1 + \cos^{2} (x) + \cos (x) \sin (x)$ in a factored form.

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Step 2.6.14.1

Rearrange terms.

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

Step 2.6.14.2

Apply pythagorean identity.

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

Step 2.6.14.3

Add $- 1$ and $1$ .

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

Step 2.6.14.4

Add $- \sin^{3} (x) \cos (x)$ and $0$ .

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

Step 2.6.14.5

Rewrite $- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) + \cos (x) \sin (x)$ in a factored form.

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Step 2.6.14.5.1

Factor $\sin (x) \cos (x)$ out of $- \sin^{3} (x) \cos (x) - \cos^{3} (x) \sin (x) + \cos (x) \sin (x)$ .

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Step 2.6.14.5.1.1

Factor $\sin (x) \cos (x)$ out of $- \sin^{3} (x) \cos (x)$ .

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

Step 2.6.14.5.1.2

Factor $\sin (x) \cos (x)$ out of $- \cos^{3} (x) \sin (x)$ .

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

Step 2.6.14.5.1.3

Factor $\sin (x) \cos (x)$ out of $\cos (x) \sin (x)$ .

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

Step 2.6.14.5.1.4

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

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

Step 2.6.14.5.1.5

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

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

Step 2.6.14.5.2

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

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

Step 2.6.14.5.3

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

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

Step 2.6.14.5.4

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

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

Step 2.6.14.5.5

Apply pythagorean identity.

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

Step 2.6.14.5.6

Multiply $- 1$ by $1$ .

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

Step 2.6.15

Add $- 1$ and $1$ .

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

Step 2.6.16

Combine exponents.

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Step 2.6.16.1

Multiply $0$ by $\sin (x)$ .

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

Step 2.6.16.2

Multiply $0$ by $\cos (x)$ .

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

Step 2.7

Divide $0$ by $(1 - 2 \cos^{2} (x)) (\sin (x) - \cos (x))$ .

$0 = 0$

Step 3

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$