Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

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

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

Step 1.3.1.4.5

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

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

Step 1.3.1.4.6

Add $1$ and $1$ .

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

Step 1.3.2

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

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

Step 1.4

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

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Step 1.4.1

Rearrange terms.

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

Step 1.4.2

Apply pythagorean identity.

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

Step 1.4.3

Add $1$ and $1$ .

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

Step 1.4.4

Factor $2$ out of $2 - 2 \cos (x)$ .

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Step 1.4.4.1

Factor $2$ out of $2$ .

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

Step 1.4.4.2

Factor $2$ out of $- 2 \cos (x)$ .

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

Step 1.4.4.3

Factor $2$ out of $2 (1) + 2 (- \cos (x))$ .

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

Step 2

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

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $2$ by $1$ .

$2$