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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2021 · Mathematics · Inverse Trigonometric Functions

What is the value of the following? [ ^ -1 (3)/(5)+ ^ -1 (3)/(2)]

What is the value of the following? $\cot \left[\sin ^{-1} \frac{3}{5}+\cot ^{-1} \frac{3}{2}\right]

Options

  1. A617\frac{6}{17}
  2. B716\frac{7}{16}
  3. C167\frac{16}{7}
  4. D176\frac{17}{6}

Correct answer

A. 617\frac{6}{17}

Step-by-step solution

 Let sin⁡−135=θ⇒sin⁡θ=35⇒tan⁡θ=352−32=34∴cot⁡=[tan⁡−134+tan⁡−123]=cot⁡[tan⁡−1(34+231−34⋅23)]=cot⁡[tan⁡−1(176)]=cot⁡[cot⁡−1(617)]=617\begin{aligned} & \text{ Let } \sin ^{-1} \frac{3}{5}=\theta \Rightarrow \sin \theta=\frac{3}{5} \\ & \Rightarrow \tan \theta=\frac{3}{\sqrt{5^2-3^2}}=\frac{3}{4} \\ & \therefore \cot =\left[\tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{2}{3}\right] \\ & =\cot \left[\tan ^{-1}\left(\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4} \cdot \frac{2}{3}}\right)\right] \\ & =\cot \left[\tan ^{-1}\left(\frac{17}{6}\right)\right]=\cot \left[\cot ^{-1}\left(\frac{6}{17}\right)\right]=\frac{6}{17} \end{aligned}
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