What is the general solution for cos(5x)-cos(x)=2sin(2x) ?

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

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cos(5x)-cos(x)=2sin(2x)

cos (5 x) - cos (x) = 2 sin (2 x)

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

Solution steps

cos (5 x) - cos (x) = 2 sin (2 x)

Subtract

2 sin (2 x)

from both sides

cos (5 x) - cos (x) - 2 sin (2 x) = 0

Rewrite using trig identities

- 2 sin (2 x) - 2 sin (2 x) sin (3 x) = 0

Factor

- 2 sin (2 x) - 2 sin (2 x) sin (3 x) : - 2 sin (2 x) (sin (3 x) + 1)

- 2 sin (2 x) (sin (3 x) + 1) = 0

Solving each part separately

sin (2 x) = 0 or sin (3 x) + 1 = 0

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

sin (3 x) + 1 = 0 : x = \frac{π}{2} + \frac{2 πn}{3}

Combine all the solutions

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

6 arccos (4 x) = 5 π

2 sin^{2} (x) - cos (x) = 1, 0 \leq x \leq 2 π

cos (θ) = - \frac{π}{2}

4 sin (t^{2}) - 1 = 1

sec (x) + 4 = 4 sec (x)

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