Trigonometry Examples

$\frac{\sin (a)}{1 + \cos (a)} + \frac{1 + \cos (a)}{\sin (a)}$

Step 1

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

$\frac{\sin (a)}{1 + \cos (a)} \cdot \frac{\sin (a)}{\sin (a)} + \frac{1 + \cos (a)}{\sin (a)}$

Step 2

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

$\frac{\sin (a)}{1 + \cos (a)} \cdot \frac{\sin (a)}{\sin (a)} + \frac{1 + \cos (a)}{\sin (a)} \cdot \frac{1 + \cos (a)}{1 + \cos (a)}$

Step 3

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

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Step 3.1

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

$\frac{\sin (a) \sin (a)}{(1 + \cos (a)) \sin (a)} + \frac{1 + \cos (a)}{\sin (a)} \cdot \frac{1 + \cos (a)}{1 + \cos (a)}$

Step 3.2

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

$\frac{\sin (a) \sin (a)}{(1 + \cos (a)) \sin (a)} + \frac{(1 + \cos (a)) (1 + \cos (a))}{\sin (a) (1 + \cos (a))}$

Step 3.3

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

$\frac{\sin (a) \sin (a)}{\sin (a) (1 + \cos (a))} + \frac{(1 + \cos (a)) (1 + \cos (a))}{\sin (a) (1 + \cos (a))}$

Step 4

Combine the numerators over the common denominator.

$\frac{\sin (a) \sin (a) + (1 + \cos (a)) (1 + \cos (a))}{\sin (a) (1 + \cos (a))}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $\sin (a) \sin (a)$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Add $1$ and $1$ .

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

Step 5.2

Expand $(1 + \cos (a)) (1 + \cos (a))$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $1$ by $1$ .

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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

Step 5.3.1.4

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

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Step 5.3.1.4.1

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

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

Step 5.3.1.4.2

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

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

Step 5.3.1.4.3

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

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

Step 5.3.1.4.4

Add $1$ and $1$ .

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

Step 5.3.2

Add $\cos (a)$ and $\cos (a)$ .

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

Step 5.4

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

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Step 5.4.1

Regroup terms.

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

Step 5.4.2

Apply pythagorean identity.

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

Step 5.4.3

Add $1$ and $1$ .

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

Step 5.4.4

Factor $2$ out of $2 + 2 \cos (a)$ .

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Step 5.4.4.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 + 2 \cos (a)}{\sin (a) (1 + \cos (a))}$

Step 5.4.4.2

Factor $2$ out of $2 \cdot 1 + 2 \cos (a)$ .

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{2}{\sin (a)}$

Step 7

Separate fractions.

$\frac{2}{1} \cdot \frac{1}{\sin (a)}$

Step 8

Convert from $\frac{1}{\sin (a)}$ to $\csc (a)$ .

$\frac{2}{1} \csc (a)$

Step 9

Divide $2$ by $1$ .

$2 \csc (a)$