Precalculus Examples

${(\csc (x) - \cot (x))}^{2}$

Step 1

Rewrite ${(\csc (x) - \cot (x))}^{2}$ as $(\csc (x) - \cot (x)) (\csc (x) - \cot (x))$ .

$(\csc (x) - \cot (x)) (\csc (x) - \cot (x))$

Step 2

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

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Step 2.1

Apply the distributive property.

$\csc (x) (\csc (x) - \cot (x)) - \cot (x) (\csc (x) - \cot (x))$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\csc (x) \csc (x)$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\csc^{2} (x) - \csc (x) \cot (x) - \cot (x) \csc (x) - \cot (x) (- \cot (x))$

Step 3.1.3

Multiply $- \cot (x) (- \cot (x))$ .

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

Multiply $\cot (x)$ by $1$ .

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Add $1$ and $1$ .

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

Step 3.2

Reorder the factors of $- \csc (x) \cot (x)$ .

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

Step 3.3

Subtract $\cot (x) \csc (x)$ from $- \cot (x) \csc (x)$ .

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