Precalculus Examples

$(1 - \cos (θ)) (\csc (θ) + \cot (θ))$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\csc (θ)$ in terms of sines and cosines.

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

Step 1.2

Rewrite $\cot (θ)$ in terms of sines and cosines.

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

Step 2

Expand $(1 - \cos (θ)) (\frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Combine $\frac{1}{\sin (θ)}$ and $\cos (θ)$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

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

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

Step 3.1.4.5

Add $1$ and $1$ .

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

Step 3.2

Subtract $\frac{\cos (θ)}{\sin (θ)}$ from $\frac{\cos (θ)}{\sin (θ)}$ .

$\frac{1}{\sin (θ)} + 0 - \frac{\cos^{2} (θ)}{\sin (θ)}$

Step 3.3

Add $\frac{1}{\sin (θ)}$ and $0$ .

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

Step 4

Combine the numerators over the common denominator.

$\frac{1 - \cos^{2} (θ)}{\sin (θ)}$

Step 5

Apply pythagorean identity.

$\frac{\sin^{2} (θ)}{\sin (θ)}$

Step 6

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

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Step 6.1

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

$\frac{\sin (θ) \sin (θ)}{\sin (θ)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Multiply by $1$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

$\frac{\sin (θ)}{1}$

Step 6.2.4

Divide $\sin (θ)$ by $1$ .

$\sin (θ)$