Solve for x (csc(x)^2)/4=4sin(x)^2

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

Multiply each term in $\frac{\csc^{2} (x)}{4} = 4 \sin^{2} (x)$ by $\frac{4}{\csc^{2} (x)}$ to eliminate the fractions.

Multiply each term in $\frac{\csc^{2} (x)}{4} = 4 \sin^{2} (x)$ by $\frac{4}{\csc^{2} (x)}$ .

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

Simplify the left side.

Combine.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$1 = 4 \sin^{2} (x) \frac{4}{\csc^{2} (x)}$

Simplify the right side.

Simplify the denominator.

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

$1 = 4 \sin^{2} (x) \frac{4}{{(\frac{1}{\sin (x)})}^{2}}$

Apply the product rule to $\frac{1}{\sin (x)}$ .

$1 = 4 \sin^{2} (x) \frac{4}{\frac{1^{2}}{\sin^{2} (x)}}$

One to any power is one.

$1 = 4 \sin^{2} (x) \frac{4}{\frac{1}{\sin^{2} (x)}}$

Multiply the numerator by the reciprocal of the denominator.

$1 = 4 \sin^{2} (x) (4 \sin^{2} (x))$

Rewrite using the commutative property of multiplication.

$1 = 4 \cdot 4 \sin^{2} (x) \sin^{2} (x)$

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

Move $\sin^{2} (x)$ .

$1 = 4 \cdot 4 (\sin^{2} (x) \sin^{2} (x))$

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

$1 = 4 \cdot 4 {\sin (x)}^{2 + 2}$

Add $2$ and $2$ .

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

Multiply $4$ by $4$ .

$1 = 16 \sin^{4} (x)$

Rewrite the equation as $16 \sin^{4} (x) = 1$ .

$16 \sin^{4} (x) = 1$

Divide each term in $16 \sin^{4} (x) = 1$ by $16$ and simplify.

Divide each term in $16 \sin^{4} (x) = 1$ by $16$ .

$\frac{16 \sin^{4} (x)}{16} = \frac{1}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 \sin^{4} (x)}{16} = \frac{1}{16}$

Divide $\sin^{4} (x)$ by $1$ .

$\sin^{4} (x) = \frac{1}{16}$

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt[4]{\frac{1}{16}}$

Simplify $\pm \sqrt[4]{\frac{1}{16}}$ .

Rewrite $\sqrt[4]{\frac{1}{16}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{16}}$ .

$\sin (x) = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{16}}$

Any root of $1$ is $1$ .

$\sin (x) = \pm \frac{1}{\sqrt[4]{16}}$

Simplify the denominator.

Rewrite $16$ as $2^{4}$ .

$\sin (x) = \pm \frac{1}{\sqrt[4]{2^{4}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\sin (x) = \pm \frac{1}{2}$

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

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

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

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

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

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

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

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

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

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

Solve for $x$ in $\sin (x) = \frac{1}{2}$ .

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

$x = \arcsin (\frac{1}{2})$

Simplify the right side.

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

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.

$x = π - \frac{π}{6}$

Simplify $π - \frac{π}{6}$ .

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Combine fractions.

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Simplify the numerator.

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

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.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Solve for $x$ in $\sin (x) = - \frac{1}{2}$ .

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

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

Simplify the right side.

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

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.

$x = 2 π + \frac{π}{6} + π$

Simplify the expression to find the second solution.

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

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 $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Combine fractions.

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Simplify the numerator.

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

List the new angles.

$x = \frac{11 π}{6}$

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate the solutions.

Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate $\frac{5 π}{6} + 2 π n$ and $\frac{11 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Solve for x (csc(x)^2)/4=4sin(x)^2

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