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

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

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cos(2x)+2cos(x)=-1

cos (2 x) + 2 cos (x) = - 1

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

Solution steps

cos (2 x) + 2 cos (x) = - 1

Subtract

- 1

from both sides

cos (2 x) + 2 cos (x) + 1 = 0

Rewrite using trig identities

2 cos (x) + 2 cos^{2} (x) = 0

Solve by substitution

cos (x) = 0, cos (x) = - 1

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = - 1 : x = π + 2 πn

Combine all the solutions

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

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

3 cot (x) = 1

tan (x) = \frac{5}{12}

cos (x) = - \frac{3}{5}

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

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