θcot θ27. Prove that fan-I + sec θ.cosecθ+I-co18 1-tan θwhereis a

1.	θcot θ27. Prove that fan-I + sec θ.cosecθ+I-co18 1-tan θwhereis an acute angle.
Answer» GIVEN:- tanθ/(1 - cotθ) + cotθ/(1 - tanθ) => tanθ/(1 - 1/tanθ) + (1/tanθ)/(1 - tanθ) => tan²θ/(tanθ - 1) - 1/tanθ(tanθ - 1) => 1/(tanθ - 1) { tan²θ - 1/tanθ } => 1/(tanθ - 1) { (tan³θ - 1)/tanθ) => {(tanθ - 1)(tan²θ + 1 + tanθ)}/{(tanθ - 1)(tanθ)} => tanθ + cotθ + 1 => sinθ/cosθ + cosθ/sinθ + 1 => (sin²θ + cos²θ)/sinθ . cosθ + 1 => 1/sinθ . cosθ + 1 => cosecθ . secθ + 1

θcot θ27. Prove that fan-I + sec θ.cosecθ+I-co18 1-tan θwhereis an acute angle.

Answer»

GIVEN:- tanθ/(1 - cotθ) + cotθ/(1 - tanθ)

=> tanθ/(1 - 1/tanθ) + (1/tanθ)/(1 - tanθ)

=> tan²θ/(tanθ - 1) - 1/tanθ(tanθ - 1)

=> 1/(tanθ - 1) { tan²θ - 1/tanθ }

=> 1/(tanθ - 1) { (tan³θ - 1)/tanθ)

=> {(tanθ - 1)(tan²θ + 1 + tanθ)}/{(tanθ - 1)(tanθ)}

=> tanθ + cotθ + 1

=> sinθ/cosθ + cosθ/sinθ + 1

=> (sin²θ + cos²θ)/sinθ . cosθ + 1

=> 1/sinθ . cosθ + 1

=> cosecθ . secθ + 1