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Question

Prove the following identities (1-16)

tan x1-cot x+cot x1-tan x=sec x cossec x+1

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Solution

LHS = tan x1-cot x+cot x1-tan x =sin xcos x1-cos xsin x+cos xsin x1-sin xcos x =sin xcos xsin x-cos xsin x+cos xsin xcos x-sin xcos x =sin xcos x×sin xsin x-cos x+cos xsin x×cos xcos x-sin x =sin xcos x×sin xsin x-cos x+cos xsin x×cos x-sin x-cos x =sin2 xcos xsin x-cos x-cos2xsin xsin x-cos x =sin3 x-cos3xsin x cos xsin x-cos x =sin x-cos xsin2 x+cos2 x+sin xcos xsin x cos xsin x-cos x =1×1+sin x cos xsin x cos x =1+sin x cos xsin x cos x =1sin x cos x+sin x cos xsin x cos x =1sin x ×1cos x+1 =cosec x ×sec x+1 =sec x cosec x+1 =RHSHence proved.

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