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JEE Main 2025 · Mathematics · Trigonometry and Inverse Trigonometric Functions · medium

The value of sin (sin ^-13/5+sin ^-18/17) is

Single correct option

The value of sin⁡ ⁣(sin⁡−135+sin⁡−1817)\sin\!\left(\sin^{-1}\dfrac35+\sin^{-1}\dfrac{8}{17}\right) is
  1. A. 8485\dfrac{84}{85}
  2. B. 3685\dfrac{36}{85}
  3. C. 1385\dfrac{13}{85}
  4. D. 7785\dfrac{77}{85}
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Answer: (D) 7785\dfrac{77}{85}

Let A=sin⁡−135A=\sin^{-1}\frac35, B=sin⁡−1817B=\sin^{-1}\frac8{17}, so cos⁡A=45\cos A=\frac45, cos⁡B=1517\cos B=\frac{15}{17}. Then sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B=35⋅1517+45⋅817=45+3285=7785\sin(A+B)=\sin A\cos B+\cos A\sin B=\frac35\cdot\frac{15}{17}+\frac45\cdot\frac8{17}=\frac{45+32}{85}=\frac{77}{85}.
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