Question

if y=(sin^-x)^2 . prove that (1-x)^2(y)n+2-(2n+1)x(y) n+2- n^2yn​

Answer 1

Answer:

Given: y=(sin

−1

x)

2

To Prove: (1×x

2

)y

2

−xy

1

−2=0

Solution: Consider y=(sin

−1

x)

2

Differentiate 'y' with respect to 'x'

dx

dy

=2sin

−1

x.

1−x

2

1

Again differentiate

dx

dy

with respect to 'x'

dx

2

d

2

y

=2

⎣

⎢

⎢

⎢

⎡

(1−x

2

)

1−x

2

1

.

1−x

2

−sin

−1

x.

2

1−x

2

1

(−2x)

⎦

⎥

⎥

⎥

⎤

dx

2

d

2

y

=

(1−x

2

)

2[1+

1−x

2

x

sin

−1

x]

Now; LHS=(1−x

2

)y

2

−xy

1

−2

=(1−x

2

).2

⎣

⎢

⎢

⎡

(1−x

2

)

1+

1−x

2

x

sin

−1

x

⎦

⎥

⎥

⎤

−

1−x

2

x(2sin

−1

x)

+(−2)

=2+

1−x

2

2x

sin

−1

x−

1−x

2

x(2sin

−1

x)

−2

=0

=RHS

Hence proved

Answer 2

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