Question

Prove the following identities ,where angles involved are acute angles gor which the expressions are defined:
I). (Cosec A-sin A) (sec A- cos A)=1/tan A+cot A.
( hint: simplify LHS and RHS separately)

Answer 1

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 proved   

Answer 2

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