Prove the following identities: `(costheta)/(1-sintheta)+(costheta)/(1+sintheta)=2sectheta`

Explanation

LHS `= frac{1+sin theta}{cos theta}+frac{cos theta}{1+sin theta}`


`=frac{(1+sin theta)^{2}+cos ^{2} theta}{cos theta(1+sin theta)}`


`= frac{1+sin ^{2} theta+2 sin theta+cos ^{2} theta}{cos theta(1+sin theta)}`


Identity : `sin ^{2} theta+cos ^{2} theta=1`


`= frac{1+1+2 sin theta}{cos theta(1+sin theta)}`


`= frac{2+2 sec theta}{cos theta(1+sin theta)}`


`= frac{2(1-1 sin theta)}{cos theta(1+sin theta)}`


`= 2 sec theta`


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