Trigonometry Examples

$(1 + \csc (x)) (1 - \csc (x))$

Step 1

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

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Step 1.1

Apply the distributive property.

$1 (1 - \csc (x)) + \csc (x) (1 - \csc (x))$

Step 1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- \csc (x)) + \csc (x) (1 - \csc (x))$

Step 1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- \csc (x)) + \csc (x) \cdot 1 + \csc (x) (- \csc (x))$

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 $1$ .

$1 + 1 (- \csc (x)) + \csc (x) \cdot 1 + \csc (x) (- \csc (x))$

Step 2.1.2

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

$1 - \csc (x) + \csc (x) \cdot 1 + \csc (x) (- \csc (x))$

Step 2.1.3

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

$1 - \csc (x) + \csc (x) + \csc (x) (- \csc (x))$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$1 - \csc (x) + \csc (x) - \csc (x) \csc (x)$

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Add $1$ and $1$ .

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

Step 2.2

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

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

Step 2.3

Add $1$ and $0$ .

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

Step 3

Simplify with factoring out.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Apply pythagorean identity.

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