Q21 / 120 · NDA 2024 (Phase 2) · Mathematics

If $4 \sin ^{-1} x+\cos ^{-1} x=\pi$, then what is $\sin ^{-1} x+4 \cos ^{-1} x$ equal to?

  • A
    $\frac{\pi}{2}$
  • B
    $\pi$
  • Correct answer
    C
    $\frac{3 \pi}{2}$
  • D
    $2 \pi$

Solution

(c) We have given, $4 \sin ^{-1} x+\cos ^{-1} x=\pi$. $\begin{aligned} & \Rightarrow 4\left(\frac{\pi}{2}-\cos ^{-1} x\right)+\left(\frac{\pi}{2}-\sin ^{-1} x\right)=\pi \\ & \left\{\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right\} \end{aligned}$ $\begin{aligned} & \Rightarrow \quad 2 \pi-4 \cos ^{-1} x+\frac{\pi}{2}-\sin ^{-1} x=\pi \\ & \Rightarrow \quad 4 \cos ^{-1} x+\sin ^{-1} x=\frac{3 \pi}{2}\end{aligned}$

Hence, the correct answer is option: 'C'.

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