Trigonometry Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Add $1$ and $1$ .

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

Step 3

Apply pythagorean identity.

$1$