Precalculus Examples

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

Step 1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \cos (x) | = | \cos (\frac{x}{2}) |$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the absolute value equation as four equations without absolute value bars.

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

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

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

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

Step 2.2

After simplifying, there are only two unique equations to be solved.

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

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

Step 2.3

Solve $\cos (x) = \cos (\frac{x}{2})$ for $x$ .

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Step 2.3.1

For the two functions to be equal, the arguments of each must be equal.

$x = \frac{x}{2}$

Step 2.3.2

Multiply both sides by $2$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the left side.

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Step 2.3.3.1.1

Move $2$ to the left of $x$ .

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

Step 2.3.3.2

Simplify the right side.

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Step 2.3.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

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

Step 2.3.3.2.1.2

Rewrite the expression.

$2 x = x$

Step 2.3.4

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.4.1

Subtract $x$ from both sides of the equation.

$2 x - x = 0$

Step 2.3.4.2

Subtract $x$ from $2 x$ .

$x = 0$

Step 2.4

List all of the solutions.

$x = 0, 0$