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

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

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

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

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

Solution steps

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

Subtract

- cos (x)

from both sides

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

Rewrite using trig identities

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

Factor

cos (x) - cos (2 x) cos (x) + 2 cos (x) sin^{2} (x) : cos (x) (1 - cos (2 x) + 2 sin^{2} (x))

cos (x) (1 - cos (2 x) + 2 sin^{2} (x)) = 0

Solving each part separately

cos (x) = 0 or 1 - cos (2 x) + 2 sin^{2} (x) = 0

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

1 - cos (2 x) + 2 sin^{2} (x) = 0 : x = πn

Combine all the solutions

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

8 tan (x) - 12 = 5 tan (x) - 9

64 = arctan (\frac{x}{11})

2 sin (3 x) cos (3 x) = 1

sin (x) = \frac{3 10}{10}

cos (θ) = 2

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