Question

cos²θ/sinθ+sinθ=cosecθ

Answer 1

L.H.S.
cos²x/sinx + sinx
(1-sin²x)/sinx+ sinx
1/sinx - sin²x/sinx + sinx
cosecx-sinx+sinx
=cosecx = R.H.S.
Hence proved

Answer 2

cos^2theta/sintheta + sintheta =cosectheta
l.h.s-
(1-sin^2theta)/sintheta + sintheta [since we know, sin^2theta+cos^2theta=1
                                         cos^2theta=1-sin^2theta]
(1-sin^2theta+sin^2theta)/sintheta
1/sintheta
cosectheta

r.h.s-
cosectheta

l.h.s=r.h.s [proved]