Answer :

Given:


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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