Prove the trigonometric identity cot 2x=csc 2x-tanx.

We'll manage the left side of the given
expression:

We know that cot (2x) = cos (2x)/sin
(2x)

We'll apply the double angle
identities:

cos (2x) = 1 - 2(sin
x)^2

sin (2x) = 2sin x*cos
x

cot (2x) = [1 - 2(sin x)^2]/sin
(2x)

[1 - 2(sin x)^2]/sin (2x) = 1/sin (2x) - 2(sin
x)^2/2sin x*cos x

But 1/sin (2x) = csc
(2x)

[1 - 2(sin x)^2]/sin (2x) = csc (2x) - sin x/cos
x

[1 - 2(sin x)^2]/sin (2x) = csc (2x) - tan
x

Since LHS = RHS, the given identity cot
(2x) = csc (2x) - tan x is verified.