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

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Q : Prove : cot 2x=cosec 2x-tanx

A : L:H:S ≡ cot 2x

= cos 2x / sin 2x

= 1- 2sin²x / sin 2x { using cos 2x = 1-2sin²x)

= 1/sin 2x - 2sin²x / sin 2x

= cosec 2x - 2sin²x / 2sinx.cosx { using sin 2x = 2sinx.cosx }

= cosec 2x - sinx/cosx

= cosec 2x - tan x

L:H:S ≡ R:H:S

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.**