What is the general solution for 2sin(x)=sqrt(cos(2x)+2) ?

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

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2sin(x)=sqrt(cos(2x)+2)

2 sin (x) = cos (2 x) + 2

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

Solution steps

2 sin (x) = cos (2 x) + 2

Subtract

cos (2 x) + 2

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combine all the solutions

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

tan (x) = \frac{7}{12}

cos (x) = 1 - sin (x)

- sin (a) - 1 = 3 sin (a) + 2

tan (x) = 7

cot (x) (tan (x) - 3) = 0

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