Precalculus Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand $(1 + \cot (x)) (1 - \cot (x))$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Apply the distributive property.

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $1$ by $1$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.1.5

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

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Step 1.1.2.1.5.1

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

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

Step 1.1.2.1.5.2

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

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

Step 1.1.2.1.5.3

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

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

Step 1.1.2.1.5.4

Add $1$ and $1$ .

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

Step 1.1.2.2

Add $- \cot (x)$ and $\cot (x)$ .

$1 + 0 - \cot^{2} (x) - \csc^{2} (x)$

Step 1.1.2.3

Add $1$ and $0$ .

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

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Move $- \csc^{2} (x)$ .

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

Step 1.2.2

Reorder $1$ and $- \csc^{2} (x)$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 2

Apply pythagorean identity.

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

Step 3

Subtract $\cot^{2} (x)$ from $- \cot^{2} (x)$ .

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