Square both sides of the equation.

Simplify each term.

Apply the sine double-angle identity.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Remove parentheses around <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mn>4</mn><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Use the double-angle identity to transform <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> to <math><mstyle displaystyle="true"><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>sin</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> .

Apply the distributive property.

Move <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify by adding terms.

Remove unnecessary parentheses.

Reorder terms.

Add <math><mstyle displaystyle="true"><mn>4</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to get <math><mstyle displaystyle="true"><mn>6</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mn>6</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><msup><mi>sin</mi><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>6</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><msup><mi>sin</mi><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mn>6</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><msup><mi>sin</mi><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Remove parentheses.

Simplify each term.

Move <math><mstyle displaystyle="true"><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mn>6</mn><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>6</mn><msup><mi>cos</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>36</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Move <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mo>-</mo><mn>12</mn></mstyle></math> .

Move <math><mstyle displaystyle="true"><msup><mi>sin</mi><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mo>-</mo><mn>12</mn></mstyle></math> .

Remove parentheses around <math><mstyle displaystyle="true"><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>sin</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Move <math><mstyle displaystyle="true"><msup><mi>sin</mi><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>12</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>sin</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>12</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>sin</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to get <math><mstyle displaystyle="true"><mo>-</mo><mn>24</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>sin</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Remove parentheses around <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Replace the <math><mstyle displaystyle="true"><msup><mi>sin</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> with <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> based on the <math><mstyle displaystyle="true"><msup><mi>sin</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mstyle></math> identity.

Move <math><mstyle displaystyle="true"><msup><mi>cos</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> .

Reorder the polynomial <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mrow><mn>6</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>24</mn><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>sin</mi><mrow><mn>4</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>4</mn><msup><mi>sin</mi><mrow><mn>6</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> alphabetically from left to right, starting with the highest order term.

Since <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> does not contain the variable to solve for, move it to the right side of the equation by subtracting <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from both sides.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Substitute <math><mstyle displaystyle="true"><mi>u</mi><mo>=</mo><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> into the equation. This will make the quadratic formula easy to use.

Since <math><mstyle displaystyle="true"><mn>4</mn><msup><mi>sin</mi><mrow><mn>6</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> does not contain the variable to solve for, move it to the right side of the equation by subtracting <math><mstyle displaystyle="true"><mn>4</mn><msup><mi>sin</mi><mrow><mn>6</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> from both sides.

Move <math><mstyle displaystyle="true"><mn>4</mn><msup><mi>sin</mi><mrow><mn>6</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the left side of the equation by adding it to both sides.

Move <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to the left side of the equation by subtracting it from both sides.

Substitute the real value of <math><mstyle displaystyle="true"><mi>u</mi><mo>=</mo><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> back into the solved equation.

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

Simplify the right side of the equation.

Rewrite <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

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

Take the inverse <math><mstyle displaystyle="true"><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>e</mi></mstyle></math> of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> from inside the <math><mstyle displaystyle="true"><mi>c</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>e</mi></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>arccos</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> to find the solution in the fourth quadrant.

To write <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Write each expression with a common denominator of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> , by multiplying each by an appropriate factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Combine.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Factor <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi><mo>⋅</mo><mn>2</mn><mo>-</mo><mi>π</mi></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify the expression.

Move <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to the left of the expression <math><mstyle displaystyle="true"><mi>π</mi><mo>⋅</mo><mn>3</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> to get <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> in the formula for period.

Solve the equation.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to get <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

The period of the <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> function is <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> so values will repeat every <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> radians in both directions.

Consolidate the answers.

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