(1/2 sin(2x))^2= 1/4 sin^2(4x)

Solution

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

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Solution steps

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

Subtract

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

from both sides

\frac{1}{4} sin^{2} (2 x) - \frac{1}{4} sin^{2} (4 x) = 0

Simplify

\frac{1}{4} sin^{2} (2 x) - \frac{1}{4} sin^{2} (4 x) : \frac{sin ^{2} ( 2 x ) - sin ^{2} ( 4 x )}{4}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

sin^{2} (2 x) - sin^{2} (4 x) : (sin (2 x) + sin (4 x)) (sin (2 x) - sin (4 x))

(sin (2 x) + sin (4 x)) (sin (2 x) - sin (4 x)) = 0

Solving each part separately

sin (2 x) + sin (4 x) = 0 or sin (2 x) - sin (4 x) = 0

sin (2 x) + sin (4 x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin (2 x) - sin (4 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = 2 πn, x = π + 2 πn

Merge Overlapping Intervals

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

8 sin^{2} (x) - 3 sin (x) - 2 = 0

2 sin^{2} (x) 3 = 7 sin (x), 0 \leq x \leq 2 π

6 sin (x) + 12 = 15

cot^{2} (x) cos (x) = cos (x)

\frac{6}{π} - 4 sin^{2} (x) = 0