What is the general solution for 8+4cos(2x)=12cos(x) ?

The general solution for 8+4cos(2x)=12cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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8+4cos(2x)=12cos(x)

8 + 4 cos (2 x) = 12 cos (x)

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

Solution steps

8 + 4 cos (2 x) = 12 cos (x)

Subtract

12 cos (x)

from both sides

8 + 4 cos (2 x) - 12 cos (x) = 0

Rewrite using trig identities

4 - 12 cos (x) + 8 cos^{2} (x) = 0

Solve by substitution

cos (x) = 1, cos (x) = \frac{1}{2}

cos (x) = 1 : x = 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

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

tan (x) = - sec (x)

3 tan^{4} (θ) + 1 = \frac{2}{tan ^{2} ( θ )}

cos^{2} (s) = \frac{3}{4}

tan (\frac{θ}{2} + \frac{π}{4}) = 1, 0 \leq θ \leq 2 π

sin (x + \frac{π}{6}) + sin (x - \frac{π}{6}) = \frac{3}{2}

What is the general solution for 8+4cos(2x)=12cos(x) ?
The general solution for 8+4cos(2x)=12cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin