Find the x-coordinates of all points on the curve
f(x) = sin 2x − 2 sin x
at which the tangent line is horizontal. (Enter your answers as a comma-separated list. Use n to represent any integer.)

Question

Find the x-coordinates of all points on the curve 
f(x) = sin 2x − 2 sin x
 at which the tangent line is horizontal. (Enter your answers as a comma-separated list. Use n to represent any integer.)

anonymous · Answer

i know f'(x) must = 0 and f'(x) = 2[cos2x-cosx]

ranga · Answer

cos2x - cosx = 0
cos2x = 2cos^2(x) - 1
Substitute.  
Let t be cosx.  You will get a quadratic equation with t.
Solve for t.  Put cos(x) back for t.
And cos(x) is a cyclical function.  So the solution will be whatever you get +/- n(2pi)

ranga · Answer

Let me put parenthesis to clarify the identity:
cos(2x) = 2cos^2(x) - 1

anonymous · Answer

$$2t^{2}-t-1=0$$

ranga · Answer

Yes.  And you can find the two roots of t.

anonymous · Answer

so solve for cosx = 1/2 and -1?

ranga · Answer

Yes.  And to each of those answers add +/-n(2pi) where n is an integer.

ranga · Answer

Did you solve for x?

ranga · Answer

Oh, when I solve for t I get 1/2 & +1.