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Question

 tan2θ(1+tan2θ)+cot2θ(1+cot2θ)=1


Solution

To prove, tan2 θ1+tan2 θ+cot2 θ1+cot2 θ=1LHS=tan2 θ1+tan2 θ+cot2 θ1+cot2 θ=tan2 θsec2 θ+cot2 θcosec2 θ=1sec2 θ×sin2 θcos2 θ+1cosec2 θ×cos2 θsin2 θcos2 θ×sin2 θcos2 θ+sin2 θ×cos2 θsin2 θ=sin2 θ+cos2 θ=1=RHS

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