Question 28 (OR 1st Question) If √(1−𝑥^2 ) + √(1−𝑦^2 ) = a (x − y), then prove that 𝑑𝑦/(𝑑𝑥 ) = √(1 − 𝑦^2 )/√(1 − 𝑥^2 ).
Finding 𝒅𝒚/𝒅𝒙 would be complicated here
To make life easy, we substitute
x = sin A
y = sin B
(As √(1−𝑥^2 )= √(1−sin^2𝐴 )=√(cos^2𝐴 ))
And then solve
Let’s substitute
x = sin A
y = sin B
in our equation
Now
√(1−𝑥^2 ) + √(1−𝑦^2 ) = a (x − y)
Putting x = sin A and y = sin B
√(1−sin^2𝐴 ) + √(1−sin^2𝐵 ) = a (sin A − sin B)
√(cos^2𝐴 ) + √(cos^2𝐵 ) = a (sin A − sin B)
cos A + cos B = a (sin A − sin B)
Using
cos A + cos B = 2 cos (𝐴+𝐵)/2 cos (𝐴−𝐵)/2
and
sin A – sin B = 2 cos (𝐴+𝐵)/2 sin (𝐴−𝐵)/2
2 cos((𝐴 + 𝐵)/2) cos((𝐴 − 𝐵)/2) = a × 2 cos((𝐴 + 𝐵)/2) s𝑖𝑛((𝐴 − 𝐵)/2)
cos((𝐴 − 𝐵)/2) = a s𝑖𝑛((𝐴 − 𝐵)/2)
〖cos 〗((𝐴 − 𝐵)/2)/〖sin 〗((𝐴 − 𝐵)/2) = a
cot((𝐴 − 𝐵)/2) = a
(𝐴 − 𝐵)/2 = 〖𝑐𝑜𝑡〗^(−1) 𝑎
𝐴−𝐵 = 2 〖𝑐𝑜𝑡〗^(−1) 𝑎
Putting back values of A and B
sin^(−1)𝑥−sin^(−1)𝑦 = 2〖𝑐𝑜𝑡〗^(−1) 𝑎
Differentiating w.r.t x
1/√(1 − 𝑥^2 )−1/√(1 − 𝑦^2 )×𝑑𝑦/𝑑𝑥=0
1/√(1 − 𝑥^2 )=1/√(1 − 𝑦^2 )×𝑑𝑦/𝑑𝑥
√(1 − 𝑦^2 )/√(1 − 𝑥^2 )=𝑑𝑦/𝑑𝑥
𝑑𝑦/(𝑑𝑥 ) = √(1 − 𝑦^2 )/√(1 − 𝑥^2 )
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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