Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

$(1 \cdot 1 + 1 \cos (θ) - \cos (θ) (1 + \cos (θ))) (1 + \cot^{2} (θ))$

Step 1.3

Apply the distributive property.

$(1 \cdot 1 + 1 \cos (θ) - \cos (θ) \cdot 1 - \cos (θ) \cos (θ)) (1 + \cot^{2} (θ))$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $1$ by $1$ .

$(1 + 1 \cos (θ) - \cos (θ) \cdot 1 - \cos (θ) \cos (θ)) (1 + \cot^{2} (θ))$

Step 2.1.2

Multiply $\cos (θ)$ by $1$ .

$(1 + \cos (θ) - \cos (θ) \cdot 1 - \cos (θ) \cos (θ)) (1 + \cot^{2} (θ))$

Step 2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.1.4

Multiply $- \cos (θ) \cos (θ)$ .

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

Add $1$ and $1$ .

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

Step 2.2

Subtract $\cos (θ)$ from $\cos (θ)$ .

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

Step 2.3

Add $1$ and $0$ .

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

Step 3

Apply pythagorean identity.

$\sin^{2} (θ) (1 + \cot^{2} (θ))$

Step 4

Apply pythagorean identity.

$\sin^{2} (θ) \csc^{2} (θ)$

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

Apply the product rule to $\frac{1}{\sin (θ)}$ .

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

Step 6.2

One to any power is one.

$\sin^{2} (θ) \frac{1}{\sin^{2} (θ)}$

Step 7

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

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Step 7.1

Cancel the common factor.

$\sin^{2} (θ) \frac{1}{\sin^{2} (θ)}$

Step 7.2

Rewrite the expression.

$1$