To prove: cot² x – tan² x = 4 cot 2x cosec 2x

Proof:

Take LHS:

cot² x – tan² x

Identities used:

a² – b² = (a – b)(a + b)

Therefore,

= (cot x – tan x)(cot x + tan x)

{∵ cot² x + 1 = cosec² x}

{∵ sin 2x = 2 sin x cos x}

= 4 cot 2x cosec 2x

= RHS

Hence Proved