Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

Add $1$ and $1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $\sin (x) (1 + \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(1 + \cos (x)) \sin (x)$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

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

Step 5.2.1.3.2

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

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

Step 5.2.1.3.3

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

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

Step 5.2.1.3.4

Add $1$ and $1$ .

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

Step 5.2.1.4

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

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Step 5.2.1.4.1

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

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

Step 5.2.1.4.2

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

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

Step 5.2.1.4.3

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

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

Step 5.2.1.4.4

Add $1$ and $1$ .

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

Step 5.2.1.5

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

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Step 5.2.1.5.1

Rearrange terms.

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

Step 5.2.1.5.2

Apply pythagorean identity.

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

Step 5.2.2

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

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Step 5.2.2.1

Reorder terms.

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.3

Combine fractions.

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Step 5.3.1

Combine the numerators over the common denominator.

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

Step 5.3.2

Reorder $- \cos^{2} (x)$ and $1$ .

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

Step 6

Apply pythagorean identity.

$\frac{\sin^{2} (x)}{\sin (x)}$

Step 7

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

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Step 7.1

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

$\frac{\sin (x) \sin (x)}{\sin (x)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Multiply by $1$ .

$\frac{\sin (x) \sin (x)}{\sin (x) \cdot 1}$

Step 7.2.2

Cancel the common factor.

$\frac{\sin (x) \sin (x)}{\sin (x) \cdot 1}$

Step 7.2.3

Rewrite the expression.

$\frac{\sin (x)}{1}$

Step 7.2.4

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

$\sin (x)$