Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $1$ by $1$ .

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Multiply $\cos (u) \cos (u)$ .

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Step 5.2.1.4.1

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

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

Step 5.2.1.4.2

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

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

Step 5.2.1.4.3

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

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

Step 5.2.1.4.4

Add $1$ and $1$ .

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

Step 5.2.2

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

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

Step 5.3

Multiply $\sin (u) \sin (u)$ .

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

Add $1$ and $1$ .

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

Step 5.4

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

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Step 5.4.1

Rearrange terms.

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

Step 5.4.2

Apply pythagorean identity.

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

Step 5.4.3

Add $1$ and $1$ .

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

Step 5.4.4

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

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Step 5.4.4.1

Factor $2$ out of $2$ .

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

Step 5.4.4.2

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

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Separate fractions.

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

Step 8

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

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

Step 9

Divide $2$ by $1$ .

$2 \csc (u)$