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Question
Prove that `tan θ/(1 - cot θ) + cot θ/(1 - tanθ)` = 1 + sec θ cosec θ
Solution
LHS = `tanθ/(1 - cot θ) + cot θ/(1 - tan θ)`
= `tan θ/(1 - 1/tanθ) + (1/tanθ)/(1 - tanθ)`
= `(tan^2θ)/(tan θ - 1) + 1/(tanθ(1 - tan θ)`
= `(tan^3θ - 1)/(tan θ(tan θ - 1))`
= `((tan θ - 1)(tan^3θ + tanθ + 1))/(tanθ(tan θ - 1))`
= `((tan^3θ + tan θ + 1))/tanθ`
= tan θ + 1 + sec
= 1 + tan θ + sec θ
= `1 + sinθ/cosθ + cosθ/sinθ`
= `1 + (sin^2θ + cos^2θ)/(sinθ cosθ)`
= `1 + 1/(sinθ cosθ)`
= 1 + sec θ cosec θ
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