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/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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

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

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

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

Subtract

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

tan (\frac{2 x}{3}) - 1 = 0

sin (θ) - cos (θ) = 1

- \frac{5}{2} csc (\frac{θ}{2}) = - 3 - csc (\frac{θ}{2})

4 cos (2 θ) + 11 sin (θ) - 4 = 3

sinh (x) = \frac{7}{24}

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/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin