Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Combine the opposite terms in $\cos (t) \cos (t) + \cos (t) \cdot 1 - 1 \cos (t) - 1 \cdot 1$ .

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Step 5.2.1

Reorder the factors in the terms $\cos (t) \cdot 1$ and $- 1 \cos (t)$ .

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

Step 5.2.2

Subtract $1 \cos (t)$ from $1 \cos (t)$ .

$\frac{\cos (t) \cos (t) + 0 - 1 \cdot 1 + \sin (t) \sin (t)}{\sin (t) (\cos (t) + 1)}$

Step 5.2.3

Add $\cos (t) \cos (t)$ and $0$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $\cos (t) \cos (t)$ .

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Step 5.3.1.1

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

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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

Step 5.3.1.4

Add $1$ and $1$ .

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

Step 5.3.2

Multiply $- 1$ by $1$ .

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

Step 5.4

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

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

Step 5.5

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 5.6

Factor $- 1$ out of $\cos^{2} (t)$ .

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

Step 5.7

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (t))$ .

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

Step 5.8

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

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

Step 5.9

Apply pythagorean identity.

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

Step 5.10

Multiply $\sin (t) \sin (t)$ .

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Step 5.10.1

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

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

Step 5.10.2

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

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

Step 5.10.3

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

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

Step 5.10.4

Add $1$ and $1$ .

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

Step 5.11

Add $- \sin^{2} (t)$ and $\sin^{2} (t)$ .

$\frac{0}{\sin (t) (\cos (t) + 1)}$

Step 6

Separate fractions.

$\frac{0}{\cos (t) + 1} \cdot \frac{1}{\sin (t)}$

Step 7

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

$\frac{0}{\cos (t) + 1} \csc (t)$

Step 8

Divide $0$ by $\cos (t) + 1$ .

$0 \csc (t)$

Step 9

Multiply $0$ by $\csc (t)$ .

$0$