Question

Prove the Identities

Tanθ/1-cotθ + cotθ/1-tanθ =1+ secθ cosecθ​

Answer 1

Answer:

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

Step-by-step explanation: