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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2023 · Mathematics · Inverse Trigonometric Functions

What is 2 ((1)/(2) ^ -1 5 3 )equal to?

What is 2cot⁡(12cos⁡−153)equalto?2 \cot \left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)equal to?

Options

  1. A-1
  2. B1
  3. C3+53+\sqrt{5}
  4. D3−53-\sqrt{5}

Correct answer

C. 3+53+\sqrt{5}

Step-by-step solution

Let, 12cos⁡−1(53)=θ⇒cos⁡2θ=53\frac{1}{2} \cos ^{-1}\left(\frac{\sqrt{5}}{3}\right)=\theta \Rightarrow \cos 2 \theta=\frac{\sqrt{5}}{3} Now, 2cot⁡θ=21+cos⁡2θ1−cos⁡2θ21+531−53=23+53−5=3+52 \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}
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