Trigonometry Examples

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

Step 1

Subtract $\cos (x)$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine $- \cos (x)$ and $\frac{1 + \cos (x)}{1 + \cos (x)}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Apply the distributive property.

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

Step 2.4.1.2

Multiply $- 1$ by $1$ .

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

Step 2.4.1.3

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

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Step 2.4.1.3.1

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

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

Step 2.4.1.3.2

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

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

Step 2.4.1.3.3

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

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

Step 2.4.1.3.4

Add $1$ and $1$ .

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

Step 2.4.1.4

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

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

Step 2.4.1.5

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

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

Step 2.4.1.6

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

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

Step 2.4.1.7

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

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

Step 2.4.1.8

Apply pythagorean identity.

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

Step 2.4.1.9

Multiply $- 1$ by $1$ .

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

Step 2.4.1.10

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

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Step 2.4.1.10.1

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

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

Step 2.4.1.10.2

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

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

Step 2.4.1.10.3

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

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

Step 2.4.1.10.4

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

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

Step 2.4.2

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

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Step 2.4.2.1

Cancel the common factor.

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

Step 2.4.2.2

Divide $- 1$ by $1$ .

$1 - 1 = 0$

Step 2.5

Subtract $1$ from $1$ .

$0 = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.