Trigonometry Examples

$(2 \csc (x) + 2) (2 \csc (x) - 2)$

Step 1

Expand $(2 \csc (x) + 2) (2 \csc (x) - 2)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$2 \csc (x) (2 \csc (x) - 2) + 2 (2 \csc (x) - 2)$

Step 1.2

Apply the distributive property.

$2 \csc (x) (2 \csc (x)) + 2 \csc (x) \cdot - 2 + 2 (2 \csc (x) - 2)$

Step 1.3

Apply the distributive property.

$2 \csc (x) (2 \csc (x)) + 2 \csc (x) \cdot - 2 + 2 (2 \csc (x)) + 2 \cdot - 2$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $2 \csc (x) (2 \csc (x)) + 2 \csc (x) \cdot - 2 + 2 (2 \csc (x)) + 2 \cdot - 2$ .

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Step 2.1.1

Reorder the factors in the terms $2 \csc (x) \cdot - 2$ and $2 (2 \csc (x))$ .

$2 \csc (x) (2 \csc (x)) - 2 \cdot 2 \csc (x) + 2 \cdot 2 \csc (x) + 2 \cdot - 2$

Step 2.1.2

Add $- 2 \cdot 2 \csc (x)$ and $2 \cdot 2 \csc (x)$ .

$2 \csc (x) (2 \csc (x)) + 0 + 2 \cdot - 2$

Step 2.1.3

Add $2 \csc (x) (2 \csc (x))$ and $0$ .

$2 \csc (x) (2 \csc (x)) + 2 \cdot - 2$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $2 \csc (x) (2 \csc (x))$ .

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Step 2.2.1.1

Multiply $2$ by $2$ .

$4 \csc (x) \csc (x) + 2 \cdot - 2$

Step 2.2.1.2

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

$4 (\csc^{1} (x) \csc (x)) + 2 \cdot - 2$

Step 2.2.1.3

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

$4 (\csc^{1} (x) \csc^{1} (x)) + 2 \cdot - 2$

Step 2.2.1.4

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

$4 {\csc (x)}^{1 + 1} + 2 \cdot - 2$

Step 2.2.1.5

Add $1$ and $1$ .

$4 \csc^{2} (x) + 2 \cdot - 2$

Step 2.2.2

Multiply $2$ by $- 2$ .

$4 \csc^{2} (x) - 4$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Reorder $4 \csc^{2} (x)$ and $- 4$ .

$- 4 + 4 \csc^{2} (x)$

Step 2.3.2

Factor $4$ out of $- 4$ .

$4 (- 1) + 4 \csc^{2} (x)$

Step 2.3.3

Factor $4$ out of $4 \csc^{2} (x)$ .

$4 (- 1) + 4 (\csc^{2} (x))$

Step 2.3.4

Factor $4$ out of $4 (- 1) + 4 (\csc^{2} (x))$ .

$4 (- 1 + \csc^{2} (x))$

Step 3

Rearrange terms.

$4 (\csc^{2} (x) - 1)$

Step 4

Apply pythagorean identity.

$4 \cot^{2} (x)$