1+sectheta/sectheta=sin^2theta/1-costheta

Question

establish each identity

Answer 1

Left side

(1+secθ)/secθ = (1+1/cosθ)/(1/cosθ)

multiply numerator and denominator by cosθ to get

(cosθ+1)/1=1+cosθ

 

Right side

sin²θ/(1-cosθ)=(1-cos²θ)/(1-cosθ)=((1+cosθ)(1-cosθ))/(1-cosθ)=1+cosθ

 We are done.

Answer 2

Since   sin^2(t) = 1-cos^2(t) = (1+cos(t))(1-cos(t)), we have,

 

               sin^2(t)/(1-cos(t)) = 1 + cos(t).

 

But, (1+sec(t))/sec(t) = (1+sec(t))*cos(t) = cos(t) + 1.

 

So, both sides are equal.