Question

cos A - sin A + 1
---------------------- = cosec A + cot A, using the identity cosec² A=1 + cot² A
cos A + sin A - 1

Answer 1

To prove that:

cos A - sin A +1

_______________= (cosec A + cot A)²

cos A + sin A -1

LHS

cos A - sin A +1

_______________

cos A + sin A -1

Dividing by sin A in both numerator and denominator,we get:

cot A - 1 + cosec A

→ __________________

cot A + 1 - cosec A

Using the identity,

cosec²A - cot²A=1→(cosec A + cot A)(cosec A - cot A)=1

cot A + cosec A -(cosec A - cot A)(cosec A + cot A)

→_________________________________________________

cot A -cosec A + (cosec A + cot A)(cosec A - cot A)

(cot A + cosec A)[1-(cosec A - cot A)]

→ ___________________________________

- (cosec A cot A)[1-(cosec A -cot A)]

cosec A + cot A

→ _______________

cosec A - cot A

Multiplying and Dividing by cosec A + cot A,

→(cosec A + cot A)²

→RHS

Hence,proved