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

The general solution for 2-cos^2(x)-sin(x)-1=cos^2(x) is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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

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

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

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

2 cos (2 x) + 1 = 0, 0 \leq x \leq 2 π

sec (x) + tan (x) - 1 = 0

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

x, tan^{2} (x) = sec (x) - 1

csc^{2} (\frac{t}{2}) = 2

What is the general solution for 2-cos^2(x)-sin(x)-1=cos^2(x) ?
The general solution for 2-cos^2(x)-sin(x)-1=cos^2(x) is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin