Question

prove that
sin theta/sin(90-theta)=sec theta cosec theta.

Answer 1

sin θ/sin(90- θ) =sin θ/cos θ ....................sin(90- θ)=cos θ
cos θ/cos(90- θ)=cos θ/sin θ.........................cos(90- θ)=sin θ
adding both of these equation and taking lcm 
we get
 sin^2 θ+cos^2 θ/sin θcos θ
and we know that 
 sin^2 θ+cos^2 θ=1
now we have
1/sin θcos θ=sec θcosec θ
taking rhs we get
sec θ=1/cos θ             and  cosec θ=1/sin θ
so
1/sin θcos θ=sec θcosec θ
hence proved