What is the general solution for cos(4x)=2cos(2x)-1 ?

The general solution for cos(4x)=2cos(2x)-1 is x=pin,x= pi/4+pin,x=(3pi)/4+pin

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cos(4x)=2cos(2x)-1

cos (4 x) = 2 cos (2 x) - 1

x = πn, x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Solution steps

cos (4 x) = 2 cos (2 x) - 1

Subtract

2 cos (2 x) - 1

from both sides

cos (4 x) - 2 cos (2 x) + 1 = 0

Rewrite using trig identities

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

Solve by substitution

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

cos (2 x) = 1 : x = πn

cos (2 x) = 0 : x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Combine all the solutions

x = πn, x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

cot^{2} (θ) + cot (θ) = 0

3 cos^{2} (θ) - cos (θ) - 4 = 0

sin (x) = \frac{120 \cdot sin ( 8 )}{30}

sin (θ) = (5)

cos^{2} (x) - 3 cos (x) - 2 = 0

What is the general solution for cos(4x)=2cos(2x)-1 ?
The general solution for cos(4x)=2cos(2x)-1 is x=pin,x= pi/4+pin,x=(3pi)/4+pin