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

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

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

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

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

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

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

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

Rewrite using trig identities

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

Factor

2 cos^{2} (x) - 2 cos (x) sin (x) : 2 cos (x) (cos (x) - sin (x))

2 cos (x) (cos (x) - sin (x)) = 0

Solving each part separately

cos (x) = 0 or cos (x) - sin (x) = 0

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

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

Combine all the solutions

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

3 cot (θ) + 5 = 0

x, h = g

9 cos (2 x) = 10 sin (x) + 1

tan^{2} (x) - 1 = 0, 0 \leq x \leq \frac{π}{2}

10 cos (B) + 6 = 4 cos (B) + 4

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