Left Hand Side:
(1+sin theta)/cos theta + cos theta/(1+sin theta)=((1+sin theta)(1+sin theta)+cos theta cos theta)/(cos theta(1+sin theta))
=(1+2sin theta + sin^2 theta + cos ^2 theta)/(cos theta(1+sin theta))
=(1+2sin theta + sin^2 theta + 1-sin^2 theta)/(cos theta(1+sin theta))
=(1+2sin theta + cancel(sin^2 theta) + 1-cancel(sin^2 theta))/(cos theta(1+sin theta))
=(2+2sin theta)/(cos theta(1+sin theta))
=(2(1+sin theta))/(cos theta(1+sin theta))
=(2 (cancel (1+sin theta)))/(cos theta cancel((1+sin theta)))
=2/cos theta
=2*1/cos theta
=2sec theta
:.= Right Hand Side