Find all points on the graph of the function

Question

f(x) = 2 cos x + cos^2 x
at which the tangent line is horizontal. (Use n as your arbitrary integer.)

smaller y-value (x,y)=

larger y-value (x,y)=

Answer 1

Since the derivative is the slope of the tangent line, we need

-2sinx - 2cosx*sinx = 0
-2sinx(1+cosx) = 0

sinx = 0 means x = nπ
cosx = -1 means x = (2n+1)π

so, f(x) has a horizontal tangent at x=nπ