Question

if secant theta+tan theta=x then prove that (x²+1) sin theta=x²–1​

Answer 1

Given:- x=secθ+tanθ

To prove:- x2+1x2−1=sinθ

Proof:-

x=secθ+tanθ

⇒x=cosθ1+sinθ

Squaring both sides, we get

⇒x2=cos2θ(1+sinθ)2

⇒x2=1−sin2θ(1+sinθ)2

⇒x2=(1+sinθ)(1−sinθ)(1+sinθ)2

⇒x2=1−sinθ(1+sinθ)

Therefore,

x2+1x2−1

=1−sinθ(1+sinθ)+11−sinθ(1+sinθ)−1

=1+sinθ+1−sinθ1+sinθ−1+sinθ

=22sinθ=sinθ

Hence proved.