Question

tanθ/1-cotθ+cotθ/1-tanθ =1 +tanθ+cotθ=1+secθ.cosecθ,Prove it by using trigonometric identities.

Answer 1

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

Answer 2

HELLO DEAR,

your questions is------------> Tanθ/1-cotθ+cotθ/1-tanθ =1 +tanθ+cotθ=1+secθ.cosecθ,Prove it by using trigonometric identities.

NOW, 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

HENCE, L.H.S = R.H.S


I HOPE ITS HELP YOU DEAR,
THANKS