Answer :


Put x = sin α and y = sin β …(i)

Now, we know that sin2θ + cos2 θ = 1

cos α + cos β = a( sin α – sin β) …(ii)

Now, we use some trigonometry formulas,

So, eq.(ii) become

2 cot-1 a = α – β …(iii)

Now, from eq.(i), we have

α = sin-1 x and β = sin-1 y

Now, put value of α and β in eq. (iii), we get

2 cot-1 a = sin-1 x – sin-1 y

or sin-1 x – sin-1 y = 2cot-1 a

On differentiating above with respect to x, we get

Hence Proved

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