Prove the following identities: `(sintheta)/(cottheta+cos e ctheta)=2+(sintheta)/(cottheta-cos e ctheta)`

Explanation

LHS` = frac{sin theta}{cot +cosec theta}=2+frac{sin theta}{cot theta-cosec theta} `


We can write expression as follow:


`=frac{sin theta}{cot theta+cosec theta}-frac{sin theta}{cot theta-cosec theta}=2`


Now LHS is,


LHS `=frac{sin theta}{(cot theta+cosec theta)}-frac{sin theta}{(cot theta-cosec theta)}`


`=sin theta{frac{(cot theta-cosec theta)-(cot theta+cos e c theta)}{(cot theta+cos e c theta)(cot theta-cosec theta)}} `


`=sin theta{frac{-2 cos e c theta}{-1}}` ….. ` (As cosec^{2} theta-cot ^{2} theta=1)`


`=sin theta times 2 cosec theta`


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