Question

Value of \( \left(\sin ^{-1} x \right)^{3}+\left(\tan ^{-1} x \right)^{3} \) can be equal to

(A) \( \frac{9 \pi^{3}}{64} \)

(B) \( \frac{\pi^{3}}{32} \)

(C) \( -\frac{\pi^{3}}{7} \)

(D) \( -\frac{3 \pi^{3}}{22} \)

Answer 1

The correct option is (B) \(\frac{\pi^3}{32}\)

Lets sin^-1 x = a and tan^-1 x = b

We know that:

a³ + b³ = (a + b)(a² − ab + b²)

1. For a = sin^-1 x:

sin a = x

2. For b = tan^-1 x:

tan b = x

Let's x = \(\frac{1}{\sqrt 2}\)​.

For x = \(\frac{1}{\sqrt 2}\)​, we have:

\(sin^{-1} \frac{1}{\sqrt2} = \frac{\pi}{4}\)

\(tan^{-1} \frac{1}{\sqrt2} = \frac{\pi}{4}\)

Now:

a³ + b³

= \((\frac{\pi}{4})^3 +(\frac{\pi}{4})^3\)

= \(2 \times (\frac{\pi}{4})^3\)

= \(2 \times \frac{\pi^3}{64}\)

= \(\frac{\pi^3}{32}\)