Trigonometry Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cot (x)$ and $b = \csc (- x)$ .

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

Step 2

Since $\csc (- x)$ is an odd function, rewrite $\csc (- x)$ as $- \csc (x)$ .

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

Step 3

Simplify each term.

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Step 3.1

Since $\csc (- x)$ is an odd function, rewrite $\csc (- x)$ as $- \csc (x)$ .

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

Step 3.2

Multiply $- - \csc (x)$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

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

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Simplify each term.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Add $1$ and $1$ .

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

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

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

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

Step 5.2.2.3

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

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

Step 5.2.2.4

Add $1$ and $1$ .

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

Step 5.3

Simplify with factoring out.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 6

Apply pythagorean identity.

$- 1 \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$- 1$