Find All Complex Solutions tan(x)sin(x)^2=tan(x)

$\tan (x) \sin^{2} (x) = \tan (x)$

Step 1

Simplify the left side.

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

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

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

Multiply $\frac{\sin (x)}{\cos (x)} \sin^{2} (x)$ .

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

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

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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

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

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

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

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

Add $1$ and $2$ .

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

Step 2

Simplify the right side.

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

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

Step 3

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

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

Step 4

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

Cancel the common factor.

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

Rewrite the expression.

$\sin^{3} (x) = \sin (x)$

Step 6

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

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

Step 7

Factor $\sin^{3} (x) - \sin (x)$ .

Factor $\sin (x)$ out of $\sin^{3} (x) - \sin (x)$ .

Factor $\sin (x)$ out of $\sin^{3} (x)$ .

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

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

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

Factor $\sin (x)$ out of $\sin (x) \sin^{2} (x) + \sin (x) \cdot - 1$ .

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

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

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

Factor.

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

$\sin (x) ((\sin (x) + 1) (\sin (x) - 1)) = 0$

Remove unnecessary parentheses.

$\sin (x) (\sin (x) + 1) (\sin (x) - 1) = 0$

Step 8

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

$\sin (x) = 0$

$\sin (x) + 1 = 0$

$\sin (x) - 1 = 0$

Step 9

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

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

$\sin (x) = 0$

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

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

$x = \arcsin (0)$

Simplify the right side.

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

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

$x = π - 0$

Subtract $0$ from $π$ .

$x = π$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

Step 10

Set $\sin (x) + 1$ equal to $0$ and solve for $x$ .

Set $\sin (x) + 1$ equal to $0$ .

$\sin (x) + 1 = 0$

Solve $\sin (x) + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$\sin (x) = - 1$

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

$x = \arcsin (- 1)$

Simplify the right side.

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$x = - \frac{π}{2}$

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Simplify the expression to find the second solution.

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Add $2 π$ to every negative angle to get positive angles.

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Combine fractions.

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

List the new angles.

$x = \frac{3 π}{2}$

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

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

Step 11

Set $\sin (x) - 1$ equal to $0$ and solve for $x$ .

Set $\sin (x) - 1$ equal to $0$ .

$\sin (x) - 1 = 0$

Solve $\sin (x) - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$\sin (x) = 1$

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

$x = \arcsin (1)$

Simplify the right side.

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

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

$x = π - \frac{π}{2}$

Simplify $π - \frac{π}{2}$ .

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

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

Combine fractions.

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

Step 12

The final solution is all the values that make $\sin (x) (\sin (x) + 1) (\sin (x) - 1) = 0$ true.

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

Step 13

Consolidate the answers.

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

Find All Complex Solutions tan(x)sin(x)^2=tan(x)

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