Question

(cosec A - sinA) (Sec A - cos A) =1 /tan A +cot A

Answer 1

LHS = (cosec A - sinA) (Sec A - cos A)

LHS = (1/sinA - sinA) (1/cosA - cosA)

LHS = (1-sin²A/sinA) (1-cos²A/cosA)

LHS = (cos²A/sinA)(sin²A/cosA

LHS = cosA × cosA__________(1)
_____________________________

RHS = 1/tanA+cotA

RHS = 1/(sinA/cosA + cosA/sinA

RHS = 1/(sin²A+cos²A/cosA×sinA)

RHS = 1/(1/cosA×sinA)

RHS = 1×(cosA×sinA)

RHS = cosA×sinA ________(2)
_____________________________

by comparing equation (1) and (2) we can said that ,
LHS = RHS
_________________________________________

Answer 2

Sol:
LHS = (Cosec A - Sin A) (Sec A - Cos A)

= (1/sin A - Sin A)(1/cos A - cos A)
= [(1 - sin2 A)/sin A] [(1 - cos2 A)/cos A]
= [(cos2 A)/sin A][sin2 A/cos A]
= sin A cos A

RHS = 1 / (Tan A + Cot A)
        = 1 / (sin A/cos A  + cos A/sin A)
        = 1 / [(sin2 A + cos2 A)/sin A cos A]
        = 1 / [1/sin A cos A]
        = sin A cos A

LHS = RHS
Hence proved.
hope this answer would have help you...
all the best for the board exams