What is the general solution for 4+2cos(2x)=6cos(x) ?

The general solution for 4+2cos(2x)=6cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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4+2cos(2x)=6cos(x)

4 + 2 cos (2 x) = 6 cos (x)

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

Solution steps

4 + 2 cos (2 x) = 6 cos (x)

Subtract

6 cos (x)

from both sides

4 + 2 cos (2 x) - 6 cos (x) = 0

Rewrite using trig identities

2 + 4 cos^{2} (x) - 6 cos (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

cos (x) - tan (x) cos (x) = 0

sin (θ) + cos (2 θ) = 0

3 sec (θ) - 4 = 0

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

cot (2 θ) = \frac{3}{4}

What is the general solution for 4+2cos(2x)=6cos(x) ?
The general solution for 4+2cos(2x)=6cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin