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NDA 2026
Exam
NDA
Year
2026
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2026 · Mathematics · Inverse Trigonometric Functions

What is [2 ^ -1 (1)/(2)-(π)/(4)] equal to ?

What is tan⁡[2tan⁡−112−π4]\tan\left[2\tan^{-1}\dfrac{1}{2}-\dfrac{\pi}{4}\right] equal to ?

Options

  1. A−7-7
  2. B00
  3. C15\dfrac{1}{5}
  4. D17\dfrac{1}{7}

Correct answer

D. 17\dfrac{1}{7}

Step-by-step solution

Let θ=tan⁡−112\theta = \tan^{-1}\dfrac{1}{2}, which implies tan⁡θ=12\tan\theta = \dfrac{1}{2}.

We need to evaluate tan⁡(2θ−π4)\tan\left(2\theta - \dfrac{\pi}{4}\right).

First, calculate tan⁡(2θ)\tan(2\theta) using the double angle formula:

tan⁡(2θ)=2tan⁡θ1−tan⁡2θ=2(12)1−(12)2=11−14=43\tan(2\theta) = \dfrac{2\tan\theta}{1 - \tan^2\theta} = \dfrac{2\left(\dfrac{1}{2}\right)}{1 - \left(\dfrac{1}{2}\right)^2} = \dfrac{1}{1 - \dfrac{1}{4}} = \dfrac{4}{3}

Now, apply the tangent subtraction formula:

tan⁡(2θ−π4)=tan⁡(2θ)−tan⁡(π4)1+tan⁡(2θ)tan⁡(π4)\tan\left(2\theta - \dfrac{\pi}{4}\right) = \dfrac{\tan(2\theta) - \tan\left(\dfrac{\pi}{4}\right)}{1 + \tan(2\theta)\tan\left(\dfrac{\pi}{4}\right)}

Substitute tan⁡(2θ)=43\tan(2\theta) = \dfrac{4}{3} and tan⁡(π4)=1\tan\left(\dfrac{\pi}{4}\right) = 1:

tan⁡(2θ−π4)=43−11+43(1)=1373=17\tan\left(2\theta - \dfrac{\pi}{4}\right) = \dfrac{\dfrac{4}{3} - 1}{1 + \dfrac{4}{3}(1)} = \dfrac{\dfrac{1}{3}}{\dfrac{7}{3}} = \dfrac{1}{7}

Answer: 17\dfrac{1}{7}
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