Prove that `(i) (tan A+ sin A)/(tan A- sin A)= (sec A+1)/(sec A-1) ` ` (ii) (cot A- cos A)/(cot A+ cos A)= ("cosec" A-1)/("cosec" A+1) `

Prove that `(i) (tan A+ sin A)/(tan A- sin A)= (sec A+1)/(sec A-1) ` `

1.	Prove that `(i) (tan A+ sin A)/(tan A- sin A)= (sec A+1)/(sec A-1) ` ` (ii) (cot A- cos A)/(cot A+ cos A)= ("cosec" A-1)/("cosec" A+1) `
Answer» We have `(i) " LHS " = (tan A + sin A)/(tan A - sin A)` ` = ((sin A)/(cos A)+ sin A)/((sin A)/(cos A)- sin A)=(sin A((1)/(cos A)+1))/(sin A((1)/(cos A)-1)) ` ` = (((1)/(cos A)+1))/(((1)/(cos A)-1))= (sec A+1)/(sec A-1)= " RHS. " ` ` therefore " LHS " = " RHS. " ` `(ii) " LHS " = (cot A - cos A)/(cot A + cos A) ` ` = ((cos A)/(sin A)- cos A)/((cos A)/(sin A)+ cos A)= (cos A((1)/(sin A)-1))/(cos A((1)/(sin A)+1))` ` =(((1)/(sin A)-1))/(((1)/(sin A)+1)) =(("cosec" A -1))/(("cosec" A +1))= "RHS. " ` ` therefore " LHS " = " RHS. " `

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