What is the general solution for 2-cos^2(x)=4sin^2(1/2 x) ?

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

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

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

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

Solution steps

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

Subtract

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

from both sides

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = 0, cos (x) = 2

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

cos (x) = 2 :

No Solution

Combine all the solutions

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

\frac{( tan ( x ) + 2 sin ( x ) )}{( tan ( x ) - 2 sin ( x ) )} = 3

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

sin^{2} (θ) = 2 sin (θ)

1 - sin (2 x) = 0

cos (p) - sin (p) = 1

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