Trigonometry Examples

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

Step 1

Subtract $\frac{\cot^{2} (x) - 1}{2 \cot (x)}$ from both sides of the equation.

$\cot (2 x) - \frac{\cot^{2} (x) - 1}{2 \cot (x)} = 0$

Step 2

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

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

$\cot (2 x) - \frac{\cot^{2} (x) - 1^{2}}{2 \cot (x)} = 0$

Step 2.1.2

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 $b = 1$ .

$\cot (2 x) - \frac{(\cot (x) + 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.2

To write $\cot (2 x)$ as a fraction with a common denominator, multiply by $\frac{2 \cot (x)}{2 \cot (x)}$ .

$\cot (2 x) \cdot \frac{2 \cot (x)}{2 \cot (x)} - \frac{(\cot (x) + 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.3

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

$\frac{\cot (2 x) (2 \cot (x))}{2 \cot (x)} - \frac{(\cot (x) + 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\cot (2 x) (2 \cot (x)) - (\cot (x) + 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

$\frac{2 \cdot \cot (2 x) \cot (x) - (\cot (x) + 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.5.2

Apply the distributive property.

$\frac{2 \cot (2 x) \cot (x) + (- \cot (x) - 1 \cdot 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.5.3

Multiply $- 1$ by $1$ .

$\frac{2 \cot (2 x) \cot (x) + (- \cot (x) - 1) (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.5.4

Expand $(- \cot (x) - 1) (\cot (x) - 1)$ using the FOIL Method.

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Step 2.5.4.1

Apply the distributive property.

$\frac{2 \cot (2 x) \cot (x) - \cot (x) (\cot (x) - 1) - 1 (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.5.4.2

Apply the distributive property.

$\frac{2 \cot (2 x) \cot (x) - \cot (x) \cot (x) - \cot (x) \cdot - 1 - 1 (\cot (x) - 1)}{2 \cot (x)} = 0$

Step 2.5.4.3

Apply the distributive property.

$\frac{2 \cot (2 x) \cot (x) - \cot (x) \cot (x) - \cot (x) \cdot - 1 - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5

Simplify and combine like terms.

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Step 2.5.5.1

Simplify each term.

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Step 2.5.5.1.1

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

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Step 2.5.5.1.1.1

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

$\frac{2 \cot (2 x) \cot (x) - (\cot^{1} (x) \cot (x)) - \cot (x) \cdot - 1 - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.1.2

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

$\frac{2 \cot (2 x) \cot (x) - (\cot^{1} (x) \cot^{1} (x)) - \cot (x) \cdot - 1 - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.1.3

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

$\frac{2 \cot (2 x) \cot (x) - {\cot (x)}^{1 + 1} - \cot (x) \cdot - 1 - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.1.4

Add $1$ and $1$ .

$\frac{2 \cot (2 x) \cot (x) - \cot^{2} (x) - \cot (x) \cdot - 1 - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.2

Multiply $- \cot (x) \cdot - 1$ .

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Step 2.5.5.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{2 \cot (2 x) \cot (x) - \cot^{2} (x) + 1 \cot (x) - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.2.2

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

$\frac{2 \cot (2 x) \cot (x) - \cot^{2} (x) + \cot (x) - 1 \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.3

Rewrite $- 1 \cot (x)$ as $- \cot (x)$ .

$\frac{2 \cot (2 x) \cot (x) - \cot^{2} (x) + \cot (x) - \cot (x) - 1 \cdot - 1}{2 \cot (x)} = 0$

Step 2.5.5.1.4

Multiply $- 1$ by $- 1$ .

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

Step 2.5.5.2

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

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

Step 2.5.5.3

Add $- \cot^{2} (x)$ and $0$ .

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

Step 3

Set the denominator in $\frac{2 \cot (2 x) \cot (x) - \cot^{2} (x) + 1}{2 \cot (x)}$ equal to $0$ to find where the expression is undefined.

$2 \cot (x) = 0$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $2 \cot (x) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 \cot (x) = 0$ by $2$ .

$\frac{2 \cot (x)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 \cot (x)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $\cot (x)$ by $1$ .

$\cot (x) = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$\cot (x) = 0$

Step 4.2

Take the inverse cotangent of both sides of the equation to extract $x$ from inside the cotangent.

$x = arccot (0)$

Step 4.3

Simplify the right side.

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Step 4.3.1

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 4.4

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{2}$

Step 4.5

Simplify $π + \frac{π}{2}$ .

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Step 4.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = π \cdot \frac{2}{2} + \frac{π}{2}$

Step 4.5.2

Combine fractions.

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Step 4.5.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} + \frac{π}{2}$

Step 4.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 4.5.3

Simplify the numerator.

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Step 4.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 4.5.3.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 4.6

Find the period of $\cot (x)$ .

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Step 4.6.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 4.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 4.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 4.6.4

Divide $π$ by $1$ .

$π$

Step 4.7

The period of the $\cot (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n$ , for any integer $n$

Step 4.8

Consolidate the answers.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 5

Set the argument in $\cot (2 x)$ equal to $π n$ to find where the expression is undefined.

$2 x = π n$ , for any integer $n$

Step 6

Divide each term in $2 x = π n$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = π n$ by $2$ .

$\frac{2 x}{2} = \frac{π n}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{π n}{2}$

Step 7

Set the argument in $\cot (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 8

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = \frac{π}{2} + π n, x = \frac{π n}{2}, x = π n}$ $n$ , for any integer $n$

Step 9