Trigonometry Examples

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

Step 1

Multiply both sides by $\csc (x) + \cot (x)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

Apply the distributive property.

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

Step 2.1.1.1.2

Apply the distributive property.

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

Step 2.1.1.1.3

Apply the distributive property.

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

Step 2.1.1.2

Simplify terms.

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Step 2.1.1.2.1

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 2.1.1.2.1.1

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

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

Step 2.1.1.2.1.2

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

$\csc (x) \csc (x) + 0 - \cot (x) \cot (x) = \frac{1}{\csc (x) + \cot (x)} (\csc (x) + \cot (x))$

Step 2.1.1.2.1.3

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

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

Step 2.1.1.2.2

Simplify each term.

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Step 2.1.1.2.2.1

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

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Step 2.1.1.2.2.1.1

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

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

Step 2.1.1.2.2.1.2

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

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

Step 2.1.1.2.2.1.3

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

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

Step 2.1.1.2.2.1.4

Add $1$ and $1$ .

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

Step 2.1.1.2.2.2

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

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Step 2.1.1.2.2.2.1

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

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

Step 2.1.1.2.2.2.2

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

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

Step 2.1.1.2.2.2.3

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

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

Step 2.1.1.2.2.2.4

Add $1$ and $1$ .

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

Step 2.1.1.3

Apply pythagorean identity.

$1 = \frac{1}{\csc (x) + \cot (x)} (\csc (x) + \cot (x))$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $\csc (x) + \cot (x)$ .

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Step 2.2.1.1

Cancel the common factor.

$1 = \frac{1}{\csc (x) + \cot (x)} (\csc (x) + \cot (x))$

Step 2.2.1.2

Rewrite the expression.

$1 = 1$

Step 3

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$