What is the general solution for sin(4x)cos(x)=cos(4x)sin(x) ?

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

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sin(4x)cos(x)=cos(4x)sin(x)

sin (4 x) cos (x) = cos (4 x) sin (x)

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

Solution steps

sin (4 x) cos (x) = cos (4 x) sin (x)

Subtract

cos (4 x) sin (x)

from both sides

sin (4 x) cos (x) - cos (4 x) sin (x) = 0

Rewrite using trig identities

sin (4 x - x) = 0

General solutions for

sin (4 x - x) = 0

4 x - x = 0 + 2 πn, 4 x - x = π + 2 πn

Solve

4 x - x = 0 + 2 πn : x = \frac{2 πn}{3}

Solve

4 x - x = π + 2 πn : x = \frac{π}{3} + \frac{2 πn}{3}

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

cos (θ) - cos (θ) = 0

2 sin^{2} (x) = \frac{csc ^{2} ( x )}{2}

4 sin^{2} (x) + 4 sin (x) = 1

5 cos (t) = 0

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

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