Question

Prove

cotAcosA/cotA+cosA = cot A-cosA/cotAcosA

Answer 1

L.H.S = Cot A Cos A/(Cot A + Cos A)

= [(Cos A/Sin A)Cos A]/[(Cos A/Sin A)+Cos A]

= (Cos² A/Sin A)/[(Cos A+Sin A Cos A)/Sin A]

= Cos² A/Cos A(1+Sin A)

= Cos A/(1+Sin A)

= [Cos A(1-Sin A)]/[(1+Sin A)(1-Sin A)]

[multiplying the numerator and the denominator with (1-Sin A)]

= [Cos A(1-Sin A)]/(1-Sin² A)

= (Cos A-Cos A SinA)/Cos²A

= [(Cos A-Cos ASin A)/Sin A]/(Cos² A/Sin A)

[dividing the numerator and the denominator by Sin A]

= [(Cos A/Sin A)-(Cos A Sin A/Sin A)]/(Cos A/Sin A)Cos A

= (Cot A-Cos A)/Cot A CosA

L.H.S = R.H.S

Hence Proved