Q176 / 270 · NDA 2023 (Phase 2) · Mathematics

What is $2 \cot \left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)$ equal to?

  • A
    -1
  • B
    1
  • Correct answer
    C
    $3+\sqrt{5}$
  • D
    $3-\sqrt{5}$

Solution

Let, $\frac{1}{2} \cos ^{-1}\left(\frac{\sqrt{5}}{3}\right)=\theta \Rightarrow \cos 2 \theta=\frac{\sqrt{5}}{3}$ Now, $2 \cot \theta=2 \sqrt{\frac{1+\cos 2 \theta}{1-\cos 2 \theta}}$ $2 \sqrt{\frac{1+\frac{\sqrt{5}}{3}}{1-\frac{\sqrt{5}}{3}}}=2 \sqrt{\frac{3+\sqrt{5}}{3-\sqrt{5}}}=3+\sqrt{5}$

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

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