MHT CET · Maths · Trigonometric Equations
The value of \(\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)\) at \(x=\frac{1}{5}\), where \(0 \leq \cos ^{-1} x \leq \pi\) and \(-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}\), is
- A \(\frac{\sqrt{6}}{5}\)
- B \(-\frac{\sqrt{6}}{5}\)
- C \(\frac{2 \sqrt{6}}{5}\)
- D \(-\frac{2 \sqrt{6}}{5}\)
Answer & Solution
Correct Answer
(D) \(-\frac{2 \sqrt{6}}{5}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right) \\ & =\cos \left[\left(\sin ^{-1} x+\cos ^{-1} x\right)+\cos ^{-1} x\right] \\ & =\cos \left(\frac{\pi}{2}+\cos ^{-1} x\right) \\ & =-\sin \left(\cos ^{-1} x\right) \\ & =-\sin \left(\sin ^{-1} \sqrt{\left(1-x^2\right)}\right) \\ & =-\sqrt{1-x^2} \\ & =-\sqrt{1-\left(\frac{1}{5}\right)^2} \\ & =-\sqrt{\frac{24}{25}}=-\frac{2 \sqrt{6}}{5}\end{aligned}\)
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