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Question
tanθ(1−cotθ)+cotθ(1−tanθ)=(1+secθcosecθ)
tanθ(1−cotθ)+cotθ(1−tanθ)=(1+secθcosecθ)
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Solution
tan θ1−cot θ+cot θ1−tan θ=(1+sec θ cosec θ)LHS=tan θ1−cot θ+cot θ1−tan θ=tan θ1−1tan θ+1tan θ1−tan θ=tan θtan θ−1tan θ+1tan θ(1−tan θ)=tan2 θtan θ−1+1tan θ(1−tan θ)=−tan2 θ1−tan θ+1tan θ(1−tan θ)=−tan3 θ+1tan θ(1−tan θ)=13−tan3 θtan θ(1−tan θ)=(1−tan θ)(1+tan2 θ+tan θ)tan θ(1−tan θ)=1+tan2 θ+tan θtan θ=sec2 θ+tan θtan θ=sec2 θtan θ+tan θtan θ=sec2 θ×cot θ+1=1cos2 θ×cos θsin θ+1=1cos θ×1sin θ+1=sec θ cosec θ+1=RHS
tan θ1−cot θ+cot θ1−tan θ=(1+sec θ cosec θ)LHS=tan θ1−cot θ+cot θ1−tan θ=tan θ1−1tan θ+1tan θ1−tan θ=tan θtan θ−1tan θ+1tan θ(1−tan θ)=tan2 θtan θ−1+1tan θ(1−tan θ)=−tan2 θ1−tan θ+1tan θ(1−tan θ)=−tan3 θ+1tan θ(1−tan θ)=13−tan3 θtan θ(1−tan θ)=(1−tan θ)(1+tan2 θ+tan θ)tan θ(1−tan θ)=1+tan2 θ+tan θtan θ=sec2 θ+tan θtan θ=sec2 θtan θ+tan θtan θ=sec2 θ×cot θ+1=1cos2 θ×cos θsin θ+1=1cos θ×1sin θ+1=sec θ cosec θ+1=RHS
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