tanθ/1 - cotθ + cotθ/1 – tanθ = 1 + secθcosecθ
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LHS = tanθ/(1 - cotθ) + cotθ/(1 - tanθ)
= tanθ/(1 - 1/tanθ) + (1/tanθ)/(1 - tanθ)
= tan²θ/(tanθ - 1) + 1/tanθ(1 - tanθ)
= tan³θ/(tanθ - 1) - 1/tanθ(tanθ - 1)
= (tan³θ - 1)/tanθ(tanθ - 1)
= (tanθ - 1)(tan²θ + 1 + tanθ)/tanθ(tanθ - 1)
= (tan²θ + 1 + tanθ)/tanθ
= tanθ + cotθ + 1
= sinθ/cosθ + cosθ/sinθ + 1
= (sin²θ + cos²θ)/sinθ.cosθ + 1
= secθ.cosecθ + 1
= 1 + secθ.cosecθ = RHS
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