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JEE Main 2017 · Mathematics · Trigonometry and Inverse Trigonometric Functions · medium

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

Single correct option

The value of sin⁡−135+sin⁡−145\sin^{-1}\dfrac35+\sin^{-1}\dfrac45 is
  1. A. π4\dfrac{\pi}{4}
  2. B. π3\dfrac{\pi}{3}
  3. C. π2\dfrac{\pi}{2}
  4. D. π\pi
Show answer and solution

Answer: (C) π2\dfrac{\pi}{2}

Let α=sin⁡−135\alpha=\sin^{-1}\tfrac35. Then cos⁡α=45\cos\alpha=\tfrac45, so sin⁡−145=sin⁡−1(cos⁡α)=π2−α\sin^{-1}\tfrac45=\sin^{-1}(\cos\alpha)=\dfrac{\pi}{2}-\alpha because α\alpha lies in [0,π/2][0,\pi/2]. Hence the sum is α+π2−α=π2\alpha+\dfrac\pi2-\alpha=\dfrac{\pi}{2}. This is the 3-4-5 right-triangle identity for complementary angles.
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