Find all points (i.e. all x-values) between 0 and 2pi where the line tangent to the graph of y =sinx/2+cosx is horizontal

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Have you considered the 1st Derivative?

1st derivative would be -(cot x) (cosx/-sinx) how would I get the points from that?

If that is the correct 1st derivative, which I didn't check, you determine where it is zero.

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