What is the general solution for sin^2(x)= 1/2+1/2 cos(2x) ?

The general solution for sin^2(x)= 1/2+1/2 cos(2x) is x= pi/4+pin,x=(3pi)/4+pin

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sin^2(x)= 1/2+1/2 cos(2x)

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

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Solution steps

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

Subtract

\frac{1}{2} + \frac{1}{2} cos (2 x)

from both sides

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

Simplify

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

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

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

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

Rewrite using trig identities

- 2 cos (2 x) = 0

Divide both sides by

- 2

cos (2 x) = 0

General solutions for

cos (2 x) = 0

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

Solve

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

Solve

2 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{4} + πn

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

tan (x) = \frac{8}{12}

tan (x) = \frac{8}{10}

3 cot (x) - 1 = 0

4 sin (3 θ) = 0

tan (2 θ) + 2 cos (θ) = 0, 0 \leq θ < 2 π

What is the general solution for sin^2(x)= 1/2+1/2 cos(2x) ?
The general solution for sin^2(x)= 1/2+1/2 cos(2x) is x= pi/4+pin,x=(3pi)/4+pin