Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \tan (x)$ .

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

Step 1.1.2.3

Simplify.

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Step 1.1.2.3.1

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

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

Step 1.1.2.3.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.5

Multiply $\frac{2 \sin (x)}{\cos (x)}$ by $\frac{1}{(1 + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})}$ .

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

Step 1.1.6

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Factor $\cos (x)$ out of $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Multiply $\cos (x)$ by $0$ .

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

Step 8

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

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

Step 9

Apply the distributive property.

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

Step 10

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

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Step 10.1

Combine $\sin (x)$ and $\frac{2 \sin (x)}{(1 + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})}$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

Add $1$ and $1$ .

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

Step 11

Rewrite using the commutative property of multiplication.

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

Step 12

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

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Step 12.1

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

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

Multiply $\sin (x)$ by $0$ .

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

Step 14

Replace the $2 \sin^{2} (x)$ with $2 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 15

Simplify each term.

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Step 15.1

Apply the distributive property.

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

Step 15.2

Multiply $2$ by $1$ .

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

Step 15.3

Multiply $- 1$ by $2$ .

$2 - 2 \cos^{2} (x) - \cos^{2} (x) = 0$

Step 16

Subtract $\cos^{2} (x)$ from $- 2 \cos^{2} (x)$ .

$2 - 3 \cos^{2} (x) = 0$

Step 17

Reorder the polynomial.

$- 3 \cos^{2} (x) + 2 = 0$

Step 18

Subtract $2$ from both sides of the equation.

$- 3 \cos^{2} (x) = - 2$

Step 19

Divide each term in $- 3 \cos^{2} (x) = - 2$ by $- 3$ and simplify.

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Step 19.1

Divide each term in $- 3 \cos^{2} (x) = - 2$ by $- 3$ .

$\frac{- 3 \cos^{2} (x)}{- 3} = \frac{- 2}{- 3}$

Step 19.2

Simplify the left side.

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Step 19.2.1

Cancel the common factor of $- 3$ .

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Step 19.2.1.1

Cancel the common factor.

$\frac{- 3 \cos^{2} (x)}{- 3} = \frac{- 2}{- 3}$

Step 19.2.1.2

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

$\cos^{2} (x) = \frac{- 2}{- 3}$

Step 19.3

Simplify the right side.

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Step 19.3.1

Dividing two negative values results in a positive value.

$\cos^{2} (x) = \frac{2}{3}$

Step 20

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\frac{2}{3}}$

Step 21

Simplify $\pm \sqrt{\frac{2}{3}}$ .

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Step 21.1

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{\sqrt{3}}$

Step 21.2

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 21.3

Combine and simplify the denominator.

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Step 21.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 21.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 21.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 21.3.4

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

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 21.3.5

Add $1$ and $1$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 21.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 21.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 21.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 21.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 21.3.6.4

Cancel the common factor of $2$ .

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Step 21.3.6.4.1

Cancel the common factor.

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 21.3.6.4.2

Rewrite the expression.

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Step 21.3.6.5

Evaluate the exponent.

$\cos (x) = \pm \frac{\sqrt{2} \sqrt{3}}{3}$

Step 21.4

Simplify the numerator.

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Step 21.4.1

Combine using the product rule for radicals.

$\cos (x) = \pm \frac{\sqrt{2 \cdot 3}}{3}$

Step 21.4.2

Multiply $2$ by $3$ .

$\cos (x) = \pm \frac{\sqrt{6}}{3}$

Step 22

The complete solution is the result of both the positive and negative portions of the solution.

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Step 22.1

First, use the positive value of the $\pm$ to find the first solution.

$\cos (x) = \frac{\sqrt{6}}{3}$

Step 22.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (x) = - \frac{\sqrt{6}}{3}$

Step 22.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (x) = \frac{\sqrt{6}}{3}, - \frac{\sqrt{6}}{3}$

Step 23

Set up each of the solutions to solve for $x$ .

$\cos (x) = \frac{\sqrt{6}}{3}$

$\cos (x) = - \frac{\sqrt{6}}{3}$

Step 24

Solve for $x$ in $\cos (x) = \frac{\sqrt{6}}{3}$ .

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Step 24.1

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

$x = \arccos (\frac{\sqrt{6}}{3})$

Step 24.2

Simplify the right side.

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Step 24.2.1

Evaluate $\arccos (\frac{\sqrt{6}}{3})$ .

$x = 0.6154797$

Step 24.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 (3.14159265) - 0.6154797$

Step 24.4

Solve for $x$ .

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Step 24.4.1

Remove parentheses.

$x = 2 (3.14159265) - 0.6154797$

Step 24.4.2

Simplify $2 (3.14159265) - 0.6154797$ .

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Step 24.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.6154797$

Step 24.4.2.2

Subtract $0.6154797$ from $6.2831853$ .

$x = 5.66770559$

Step 24.5

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

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Step 24.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 24.5.2

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

$\frac{2 π}{| 1 |}$

Step 24.5.3

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

$\frac{2 π}{1}$

Step 24.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 24.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.6154797 + 2 π n, 5.66770559 + 2 π n$ , for any integer $n$

Step 25

Solve for $x$ in $\cos (x) = - \frac{\sqrt{6}}{3}$ .

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Step 25.1

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

$x = \arccos (- \frac{\sqrt{6}}{3})$

Step 25.2

Simplify the right side.

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Step 25.2.1

Evaluate $\arccos (- \frac{\sqrt{6}}{3})$ .

$x = 2.52611294$

Step 25.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 2.52611294$

Step 25.4

Solve for $x$ .

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Step 25.4.1

Remove parentheses.

$x = 2 (3.14159265) - 2.52611294$

Step 25.4.2

Simplify $2 (3.14159265) - 2.52611294$ .

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Step 25.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.52611294$

Step 25.4.2.2

Subtract $2.52611294$ from $6.2831853$ .

$x = 3.75707236$

Step 25.5

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

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Step 25.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 25.5.2

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

$\frac{2 π}{| 1 |}$

Step 25.5.3

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

$\frac{2 π}{1}$

Step 25.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 25.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2.52611294 + 2 π n, 3.75707236 + 2 π n$ , for any integer $n$

Step 26

List all of the solutions.

$x = 0.6154797 + 2 π n, 5.66770559 + 2 π n, 2.52611294 + 2 π n, 3.75707236 + 2 π n$ , for any integer $n$

Step 27

Consolidate the solutions.

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Step 27.1

Consolidate $0.6154797 + 2 π n$ and $3.75707236 + 2 π n$ to $0.6154797 + π n$ .

$x = 0.6154797 + π n, 5.66770559 + 2 π n, 2.52611294 + 2 π n$ , for any integer $n$

Step 27.2

Consolidate $5.66770559 + 2 π n$ and $2.52611294 + 2 π n$ to $2.52611294 + π n$ .

$x = 0.6154797 + π n, 2.52611294 + π n$ , for any integer $n$

Step 28

Exclude the solutions that do not make $\frac{2 \tan (x)}{1 - \tan^{2} (x)} - \cot (x) = 0$ true.

No solution