TS EAMCET · Maths · Inverse Trigonometric Functions
\(\begin{aligned} & 2 \sin ^{-1} x+\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)+3 \cos ^{-1} x \\ & -\cos ^{-1}\left(4 x^3-3 x\right) \text { is equal to }\end{aligned}\)
- A \(4 \sin ^{-1} x\), when \(x \in[-1,1]\)
- B \(\pi\), when \(x \in\left[-1,-\frac{1}{\sqrt{2}}\right]\)
- C \(-\pi\), when \(x \in\left[\frac{-1}{2}, \frac{1}{2}\right]\)
- D \(4 \sin ^{-1} x+2 \cos ^{-1}\left(4 x^3-3 x\right), x \in\left[\frac{1}{\sqrt{2}}, 1\right]\)
Answer & Solution
Correct Answer
(B) \(\pi\), when \(x \in\left[-1,-\frac{1}{\sqrt{2}}\right]\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { } 2 \sin ^{-1} x+\sin ^{-1}\left[2 x \sqrt{1-x^2}\right] \\ & +3 \cos ^{-1} x-\cos ^{-1}\left(4 x^3-3 x\right) \\ & \text { Put } x=\cos \theta \Rightarrow \theta=\cos ^{-1} x \\ & =2 \sin ^{-1}(\cos \theta)+\sin ^{-1}\left(2 \cos \theta \sqrt{1-\cos ^2…
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