Question

If y = (sin⁻¹ x)², then (1 − x²)y₂ is equal to

 

Options

• xy₁ + 2

•  xy₁ − 2

• −xy₁+2

• none of these

Answer 1

(a) xy₁ + 2

Here,

\[y = \left( \sin^{- 1} x \right)^2 \]
\[\text { Now,} \]
\[ y_1 = 2 \sin^{- 1} x \frac{1}{\sqrt{1 - x^2}}\]
\[ \Rightarrow y_2 = \frac{2}{1 - x^2} + \frac{2x \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}}\]
\[ \Rightarrow y_2 = \frac{2}{1 - x^2} + \frac{2x \sin^{- 1} x}{\left( 1 - x^2 \right)\sqrt{1 - x^2}}\]
\[ \Rightarrow y_2 = \frac{2}{1 - x^2} + \frac{x y_1}{\left( 1 - x^2 \right)}\]
\[ \Rightarrow y_2 \left( 1 - x^2 \right) = 2 + x y_1\]