Question

Prove that sin theta cos theta = cot theta is not a trigonometric identify by producing a counterexample

Answer 1

One counterexample is theta = pi/4, because

sin(theta)*cos(theta) = sin(pi/4)*cos(pi/4) = sqrt(2)/2*sqrt(2)/2 = 1/2

while

cot(theta) = cot(pi/4) = cos(pi/4)/sin(pi/4) = (sqrt(2)/2)/(sqrt(2)/2) = 1

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After plugging in theta = pi/4, the equation

sin(theta)cos(theta) = cot(theta)

becomes

1/2 = 1

which is a false equation. This is enough to show we do not have an identity.