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

The general solution for sin(x)=2cos^2(x)-1 is x= pi/6+2pin,x=(5pi)/6+2pin,x=(3pi)/2+2pin

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

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solution steps

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

Subtract

2 cos^{2} (x) - 1

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

tan (t) = \frac{- 5}{12}

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

x, arctan (x) = 0

5 sin (x) = - 3

csc^{2} (x) = 2 \cdot cot^{2} (x)

What is the general solution for sin(x)=2cos^2(x)-1 ?
The general solution for sin(x)=2cos^2(x)-1 is x= pi/6+2pin,x=(5pi)/6+2pin,x=(3pi)/2+2pin