Precalculus Examples

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

Step 1

Expand $(\cot (θ) + \csc (θ)) (\cot (θ) - \csc (θ))$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\cot (θ) \cot (θ) + \cot (θ) (- \csc (θ)) + \csc (θ) \cot (θ) + \csc (θ) (- \csc (θ))$ .

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Step 2.1.1

Reorder the factors in the terms $\cot (θ) (- \csc (θ))$ and $\csc (θ) \cot (θ)$ .

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

Step 2.1.2

Add $- \cot (θ) \csc (θ)$ and $\cot (θ) \csc (θ)$ .

$\cot (θ) \cot (θ) + 0 + \csc (θ) (- \csc (θ))$

Step 2.1.3

Add $\cot (θ) \cot (θ)$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $\cot (θ) \cot (θ)$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

$\cot^{2} (θ) - \csc (θ) \csc (θ)$

Step 2.2.3

Multiply $- \csc (θ) \csc (θ)$ .

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

$\cot^{2} (θ) - {\csc (θ)}^{1 + 1}$

Step 2.2.3.4

Add $1$ and $1$ .

$\cot^{2} (θ) - \csc^{2} (θ)$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Reorder $\cot^{2} (θ)$ and $- \csc^{2} (θ)$ .

$- \csc^{2} (θ) + \cot^{2} (θ)$

Step 2.3.2

Factor $- 1$ out of $- \csc^{2} (θ)$ .

$- (\csc^{2} (θ)) + \cot^{2} (θ)$

Step 2.3.3

Factor $- 1$ out of $\cot^{2} (θ)$ .

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

Step 2.3.4

Factor $- 1$ out of $- (\csc^{2} (θ)) - 1 (- \cot^{2} (θ))$ .

$- (\csc^{2} (θ) - \cot^{2} (θ))$

Step 3

Apply pythagorean identity.

$- 1 \cdot 1$

Step 4

Multiply $- 1$ by $1$ .

$- 1$