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

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

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

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

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

Solution steps

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

Subtract

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

from both sides

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

y, cos (y) = x

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

1 + sin (x) = cos (x)

cos (2 x) = \frac{2}{π}

4 tan^{2} (x) + 8 sec (x) = 1

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