Prove the following identities: `(cotA+cos e cA-1)/(cotA-cos e cA+1)=(1+cosA)/(sinA)`

Explanation

LHS `= frac{cot A+cosec A-1}{cot A-cosec A+1} `


`=frac{cot A+cosec A-(cosec^{2} A-cot ^{2} A)}{cot A-cosec A+1}` ……`[AS cosec^{2} A-cot ^{2} A=1] `


`=frac{(cot A+cosec A)-(cot A+cosec A)(cosec A-cot A)}{cot A-cosec A+1}`


`=frac{(cot A+cosec A)(1-cosec A+cot A)}{cot A-cosec A+1} `


`cot A+cosec A=frac{cos A}{sin A}+frac{1}{sin A} `


`=frac{cos A+1}{sin A} `


`=frac{1+cos A}{sin A}`


=RHS