Trigonometry Examples

$4 \cot (x) = \cot (x) \sin^{2} (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $4 \cot (x)$ .

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Step 1.1.1

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

$4 \frac{\cos (x)}{\sin (x)} = \cot (x) \sin^{2} (x)$

Step 1.1.2

Combine $4$ and $\frac{\cos (x)}{\sin (x)}$ .

$\frac{4 \cos (x)}{\sin (x)} = \cot (x) \sin^{2} (x)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\cot (x) \sin^{2} (x)$ .

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Step 2.1.1

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

$\frac{4 \cos (x)}{\sin (x)} = \frac{\cos (x)}{\sin (x)} \sin^{2} (x)$

Step 2.1.2

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

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Step 2.1.2.1

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

$\frac{4 \cos (x)}{\sin (x)} = \frac{\cos (x)}{\sin (x)} (\sin (x) \sin (x))$

Step 2.1.2.2

Cancel the common factor.

$\frac{4 \cos (x)}{\sin (x)} = \frac{\cos (x)}{\sin (x)} (\sin (x) \sin (x))$

Step 2.1.2.3

Rewrite the expression.

$\frac{4 \cos (x)}{\sin (x)} = \cos (x) \sin (x)$

Step 3

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

$\sin (x) \frac{4 \cos (x)}{\sin (x)} = \sin (x) (\cos (x) \sin (x))$

Step 4

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

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Step 4.1

Cancel the common factor.

$\sin (x) \frac{4 \cos (x)}{\sin (x)} = \sin (x) (\cos (x) \sin (x))$

Step 4.2

Rewrite the expression.

$4 \cos (x) = \sin (x) (\cos (x) \sin (x))$

Step 5

Multiply $\sin (x) (\cos (x) \sin (x))$ .

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Step 5.1

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

$4 \cos (x) = \sin^{1} (x) \sin (x) \cos (x)$

Step 5.2

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

$4 \cos (x) = \sin^{1} (x) \sin^{1} (x) \cos (x)$

Step 5.3

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

$4 \cos (x) = {\sin (x)}^{1 + 1} \cos (x)$

Step 5.4

Add $1$ and $1$ .

$4 \cos (x) = \sin^{2} (x) \cos (x)$

Step 6

Subtract $\sin^{2} (x) \cos (x)$ from both sides of the equation.

$4 \cos (x) - \sin^{2} (x) \cos (x) = 0$

Step 7

Factor $4 \cos (x) - \sin^{2} (x) \cos (x)$ .

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Step 7.1

Factor $\cos (x)$ out of $4 \cos (x) - \sin^{2} (x) \cos (x)$ .

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Step 7.1.1

Factor $\cos (x)$ out of $4 \cos (x)$ .

$\cos (x) \cdot 4 - \sin^{2} (x) \cos (x) = 0$

Step 7.1.2

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

$\cos (x) \cdot 4 + \cos (x) (- \sin^{2} (x)) = 0$

Step 7.1.3

Factor $\cos (x)$ out of $\cos (x) \cdot 4 + \cos (x) (- \sin^{2} (x))$ .

$\cos (x) (4 - \sin^{2} (x)) = 0$

Step 7.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.3

Factor.

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Step 7.3.1

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

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

Step 7.3.2

Remove unnecessary parentheses.

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

Step 8

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

$2 + \sin (x) = 0$

$2 - \sin (x) = 0$

Step 9

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 9.2

Solve $\cos (x) = 0$ for $x$ .

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Step 9.2.1

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

$x = \arccos (0)$

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

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

$x = \frac{π}{2}$

Step 9.2.3

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.

$x = 2 π - \frac{π}{2}$

Step 9.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 9.2.4.1

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

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

Step 9.2.4.2

Combine fractions.

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Step 9.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 9.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 9.2.4.3

Simplify the numerator.

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Step 9.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 9.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 9.2.5

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

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Step 9.2.5.1

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

$\frac{2 π}{| b |}$

Step 9.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 9.2.5.3

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

$\frac{2 π}{1}$

Step 9.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.2.6

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

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

Step 10

Set $2 + \sin (x)$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $2 + \sin (x)$ equal to $0$ .

$2 + \sin (x) = 0$

Step 10.2

Solve $2 + \sin (x) = 0$ for $x$ .

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Step 10.2.1

Subtract $2$ from both sides of the equation.

$\sin (x) = - 2$

Step 10.2.2

The range of sine is $- 1 \leq y \leq 1$ . Since $- 2$ does not fall in this range, there is no solution.

No solution

Step 11

The final solution is all the values that make $\cos (x) (2 + \sin (x)) (2 - \sin (x)) = 0$ true.

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

Step 12

Consolidate the answers.

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