O lvl A Maths: Trigonometric Identity

Prove the following identity:

(tan x – cosec x)² – ( cot x – sec x)² = 2(cosec x – sec x)

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Answer:

LHS
= tan² x - 2 tan x cosec x + cosec² x - cot² x + 2 cot x sec x - sec² x
= tan² x - 2 sec x + cosec² x - cot² x + 2 cosec x - sec² x
= ( tan² x - sec² x ) + (cosec² x - cot² x) + 2 cosec x - 2 sec x
= -1 + 1 + 2 cosec x - 2 sec x
= 2 (cosec x - sec x)
= RHS (proved)