Trigonometry Examples

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

Step 1

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

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Step 1.1

Subtract $1$ from both sides of the equation.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

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.

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

Step 2.4.1.2

Multiply $- 1$ by $1$ .

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

Step 2.4.1.3

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

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Step 2.4.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.4.1.3.2

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

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

Step 2.4.1.3.3

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

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

Step 2.4.1.3.4

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

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

Step 2.4.1.3.5

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

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

Step 2.4.1.3.6

Add $1$ and $1$ .

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

Step 2.4.1.4

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

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Step 2.4.1.4.1

Rearrange terms.

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

Step 2.4.1.4.2

Apply pythagorean identity.

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

Step 2.4.2

Cancel the common factor of $- \cos (a) + 1$ and $1 - \cos (a)$ .

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Step 2.4.2.1

Reorder terms.

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

$1 - 1 = 0$

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.