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NDA 2023 Maths · Mathematics · Inverse Trigonometric Functions · hard

The value of sin ^-14/5+sin ^-15/13+sin ^-116/65 is

Single correct option

The value of sin⁡−145+sin⁡−1513+sin⁡−11665\sin^{-1}\dfrac45+\sin^{-1}\dfrac5{13}+\sin^{-1}\dfrac{16}{65} is
  1. A. π6\dfrac\pi6
  2. B. π4\dfrac\pi4
  3. C. π2\dfrac\pi2
  4. D. π\pi
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Answer: (C) π2\dfrac\pi2

Let α=sin⁡−145\alpha=\sin^{-1}\frac45, β=sin⁡−1513\beta=\sin^{-1}\frac5{13}. Then cos⁡(α+β)=35⋅1213−45⋅513=1665\cos(\alpha+\beta)=\frac35\cdot\frac{12}{13}-\frac45\cdot\frac5{13}=\frac{16}{65}, so α+β=cos⁡−11665=π2−sin⁡−11665\alpha+\beta=\cos^{-1}\frac{16}{65}=\frac\pi2-\sin^{-1}\frac{16}{65}. Hence the sum of all three is π2\frac\pi2.
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