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The proof is given here:
On the L.H.S side we've, 1/1+sin theta +1/1-sin theta
and on the R.H.S side we've 2sec²theta
so, on LHS side,
after taking LCM ,
it came out to be,
1+sin theta+1-sin theta /1-sin²theta
so, 2/ 1-sin²theta {cos² theta=1-sin²theta}
so, 2/cos²theta
and, 1/cos²theta =sec² theta
therefore, 2×sec²theta
= 2sec²theta.
now, LHS=2sec²theta
and, RHS=2sec²theta
therefore, LHS=RHS
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