Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Rewrite $\cot (2 x)$ in terms of sines and cosines.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cot^{2} (x) - 1}{2 \cot (x)}$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{\cot^{2} (x) - 1}{2 \cot (x)}$ .

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Step 2.1.1

Simplify the numerator.

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Step 2.1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cot^{2} (x) - 1^{2}}{2 \cot (x)}$

Step 2.1.1.2

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 $b = 1$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{(\cot (x) + 1) (\cot (x) - 1)}{2 \cot (x)}$

Step 2.1.1.3

Simplify.

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Step 2.1.1.3.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{(\frac{\cos (x)}{\sin (x)} + 1) (\cot (x) - 1)}{2 \cot (x)}$

Step 2.1.1.3.2

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{(\frac{\cos (x)}{\sin (x)} + 1) (\frac{\cos (x)}{\sin (x)} - 1)}{2 \cot (x)}$

Step 2.1.2

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{(\frac{\cos (x)}{\sin (x)} + 1) (\frac{\cos (x)}{\sin (x)} - 1)}{2 \frac{\cos (x)}{\sin (x)}}$

Step 2.1.3

Combine $2$ and $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{(\frac{\cos (x)}{\sin (x)} + 1) (\frac{\cos (x)}{\sin (x)} - 1)}{\frac{2 \cos (x)}{\sin (x)}}$

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos (x)}{\sin (x)} + 1) (\frac{\cos (x)}{\sin (x)} - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.5

Expand $(\frac{\cos (x)}{\sin (x)} + 1) (\frac{\cos (x)}{\sin (x)} - 1)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos (x)}{\sin (x)} (\frac{\cos (x)}{\sin (x)} - 1) + 1 (\frac{\cos (x)}{\sin (x)} - 1)) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.5.2

Apply the distributive property.

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 (\frac{\cos (x)}{\sin (x)} - 1)) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.5.3

Apply the distributive property.

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $\frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$ .

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Step 2.1.6.1.1.1

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos (x) \cos (x)}{\sin (x) \sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.2

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

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{1} (x) \cos (x)}{\sin (x) \sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.3

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

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{1} (x) \cos^{1} (x)}{\sin (x) \sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.4

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

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{{\cos (x)}^{1 + 1}}{\sin (x) \sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.5

Add $1$ and $1$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin (x) \sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.6

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

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{1} (x) \sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.7

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

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{1} (x) \sin^{1} (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.8

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

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{{\sin (x)}^{1 + 1}} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.1.9

Add $1$ and $1$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} + \frac{\cos (x)}{\sin (x)} \cdot - 1 + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.2

Combine $\frac{\cos (x)}{\sin (x)}$ and $- 1$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} + \frac{\cos (x) \cdot - 1}{\sin (x)} + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.3

Move $- 1$ to the left of $\cos (x)$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} + \frac{- 1 \cdot \cos (x)}{\sin (x)} + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.4

Move the negative in front of the fraction.

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos (x)}{\sin (x)} + 1 \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.5

Multiply $\frac{\cos (x)}{\sin (x)}$ by $1$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} + 1 \cdot - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.1.6

Multiply $- 1$ by $1$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} - \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.2

Add $- \frac{\cos (x)}{\sin (x)}$ and $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} + 0 - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.6.3

Add $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ and $0$ .

$\frac{\cos (2 x)}{\sin (2 x)} = (\frac{\cos^{2} (x)}{\sin^{2} (x)} - 1) \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.7

Simplify terms.

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Step 2.1.7.1

Apply the distributive property.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\sin (x)}{2 \cos (x)} - 1 \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.7.2

Combine.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos^{2} (x) \sin (x)}{\sin^{2} (x) (2 \cos (x))} - 1 \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.7.3

Rewrite $- 1 \frac{\sin (x)}{2 \cos (x)}$ as $- \frac{\sin (x)}{2 \cos (x)}$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos^{2} (x) \sin (x)}{\sin^{2} (x) (2 \cos (x))} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8

Simplify each term.

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Step 2.1.8.1

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 2.1.8.1.1

Factor $\cos (x)$ out of $\cos^{2} (x) \sin (x)$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos (x) (\cos (x) \sin (x))}{\sin^{2} (x) (2 \cos (x))} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.1.2

Cancel the common factors.

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Step 2.1.8.1.2.1

Factor $\cos (x)$ out of $\sin^{2} (x) (2 \cos (x))$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos (x) (\cos (x) \sin (x))}{\cos (x) (\sin^{2} (x) \cdot (2))} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.1.2.2

Cancel the common factor.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos (x) (\cos (x) \sin (x))}{\cos (x) (\sin^{2} (x) \cdot (2))} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.1.2.3

Rewrite the expression.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos (x) \sin (x)}{\sin^{2} (x) \cdot (2)} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.2

Cancel the common factor of $\sin (x)$ and $\sin^{2} (x)$ .

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Step 2.1.8.2.1

Factor $\sin (x)$ out of $\cos (x) \sin (x)$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\sin (x) \cos (x)}{\sin^{2} (x) \cdot (2)} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.2.2

Cancel the common factors.

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Step 2.1.8.2.2.1

Factor $\sin (x)$ out of $\sin^{2} (x) \cdot (2)$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\sin (x) \cos (x)}{\sin (x) (\sin (x) \cdot 2)} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.2.2.2

Cancel the common factor.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\sin (x) \cos (x)}{\sin (x) (\sin (x) \cdot 2)} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.2.2.3

Rewrite the expression.

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos (x)}{\sin (x) \cdot 2} - \frac{\sin (x)}{2 \cos (x)}$

Step 2.1.8.3

Move $2$ to the left of $\sin (x)$ .

$\frac{\cos (2 x)}{\sin (2 x)} = \frac{\cos (x)}{2 \sin (x)} - \frac{\sin (x)}{2 \cos (x)}$

Step 3

Multiply both sides of the equation by $\sin (2 x)$ .

$\sin (2 x) \frac{\cos (2 x)}{\sin (2 x)} = \sin (2 x) (\frac{\cos (x)}{2 \sin (x)} - \frac{\sin (x)}{2 \cos (x)})$

Step 4

Cancel the common factor of $\sin (2 x)$ .

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Step 4.1

Cancel the common factor.

$\sin (2 x) \frac{\cos (2 x)}{\sin (2 x)} = \sin (2 x) (\frac{\cos (x)}{2 \sin (x)} - \frac{\sin (x)}{2 \cos (x)})$

Step 4.2

Rewrite the expression.

$\cos (2 x) = \sin (2 x) (\frac{\cos (x)}{2 \sin (x)} - \frac{\sin (x)}{2 \cos (x)})$

Step 5

Apply the distributive property.

$\cos (2 x) = \sin (2 x) \frac{\cos (x)}{2 \sin (x)} + \sin (2 x) (- \frac{\sin (x)}{2 \cos (x)})$

Step 6

Combine $\sin (2 x)$ and $\frac{\cos (x)}{2 \sin (x)}$ .

$\cos (2 x) = \frac{\sin (2 x) \cos (x)}{2 \sin (x)} + \sin (2 x) (- \frac{\sin (x)}{2 \cos (x)})$

Step 7

Rewrite using the commutative property of multiplication.

$\cos (2 x) = \frac{\sin (2 x) \cos (x)}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8

Simplify each term.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the sine double-angle identity.

$\cos (2 x) = \frac{2 \sin (x) \cos (x) \cos (x)}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.1.2

Combine exponents.

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Step 8.1.2.1

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

$\cos (2 x) = \frac{2 \sin (x) (\cos^{1} (x) \cos (x))}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.1.2.2

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

$\cos (2 x) = \frac{2 \sin (x) (\cos^{1} (x) \cos^{1} (x))}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.1.2.3

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

$\cos (2 x) = \frac{2 \sin (x) {\cos (x)}^{1 + 1}}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.1.2.4

Add $1$ and $1$ .

$\cos (2 x) = \frac{2 \sin (x) \cos^{2} (x)}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Cancel the common factor.

$\cos (2 x) = \frac{2 \sin (x) \cos^{2} (x)}{2 \sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.2.2

Rewrite the expression.

$\cos (2 x) = \frac{\sin (x) \cos^{2} (x)}{\sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.3

Cancel the common factor of $\sin (x)$ .

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Step 8.3.1

Cancel the common factor.

$\cos (2 x) = \frac{\sin (x) \cos^{2} (x)}{\sin (x)} - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.3.2

Divide $\cos^{2} (x)$ by $1$ .

$\cos (2 x) = \cos^{2} (x) - \sin (2 x) \frac{\sin (x)}{2 \cos (x)}$

Step 8.4

Combine $\frac{\sin (x)}{2 \cos (x)}$ and $\sin (2 x)$ .

$\cos (2 x) = \cos^{2} (x) - \frac{\sin (x) \sin (2 x)}{2 \cos (x)}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Apply the sine double-angle identity.

$\cos (2 x) = \cos^{2} (x) - \frac{\sin (x) \cdot 2 \sin (x) \cos (x)}{2 \cos (x)}$

Step 8.5.2

Combine exponents.

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Step 8.5.2.1

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

$\cos (2 x) = \cos^{2} (x) - \frac{2 (\sin^{1} (x) \sin (x)) \cos (x)}{2 \cos (x)}$

Step 8.5.2.2

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

$\cos (2 x) = \cos^{2} (x) - \frac{2 (\sin^{1} (x) \sin^{1} (x)) \cos (x)}{2 \cos (x)}$

Step 8.5.2.3

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

$\cos (2 x) = \cos^{2} (x) - \frac{2 {\sin (x)}^{1 + 1} \cos (x)}{2 \cos (x)}$

Step 8.5.2.4

Add $1$ and $1$ .

$\cos (2 x) = \cos^{2} (x) - \frac{2 \sin^{2} (x) \cos (x)}{2 \cos (x)}$

Step 8.6

Cancel the common factor of $2$ .

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Step 8.6.1

Cancel the common factor.

$\cos (2 x) = \cos^{2} (x) - \frac{2 \sin^{2} (x) \cos (x)}{2 \cos (x)}$

Step 8.6.2

Rewrite the expression.

$\cos (2 x) = \cos^{2} (x) - \frac{\sin^{2} (x) \cos (x)}{\cos (x)}$

Step 8.7

Cancel the common factor of $\cos (x)$ .

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Step 8.7.1

Cancel the common factor.

$\cos (2 x) = \cos^{2} (x) - \frac{\sin^{2} (x) \cos (x)}{\cos (x)}$

Step 8.7.2

Divide $\sin^{2} (x)$ by $1$ .

$\cos (2 x) = \cos^{2} (x) - \sin^{2} (x)$

Step 9

Apply the cosine double-angle identity.

$\cos (2 x) = \cos (2 x)$

Step 10

Move all terms containing $\cos (2 x)$ to the left side of the equation.

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Step 10.1

Subtract $\cos (2 x)$ from both sides of the equation.

$\cos (2 x) - \cos (2 x) = 0$

Step 10.2

Subtract $\cos (2 x)$ from $\cos (2 x)$ .

$0 = 0$

Step 11

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 12

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$