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Let-f-x-sin-tan-1-x-sin-cot-1-x-2-1-where-x-gt-1-and-dy-dx-1-2-d-dx-sin-1-f-x-if-y-3-6-then-y-3-




Question Number 145814 by gsk2684 last updated on 08/Jul/21
Let f(x)={sin (tan^(−1) x)+sin (cot^(−1) x)}^2 −1  where ∣x∣>1 and (dy/dx)=(1/2)(d/dx)(sin^(−1) f(x)).  if y((√3))=(Π/6) then y(−(√3))=?
$${Let}\:{f}\left({x}\right)=\left\{\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} {x}\right)+\mathrm{sin}\:\left(\mathrm{cot}^{−\mathrm{1}} {x}\right)\right\}^{\mathrm{2}} −\mathrm{1} \\ $$$${where}\:\mid{x}\mid>\mathrm{1}\:{and}\:\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{\mathrm{2}}\frac{{d}}{{dx}}\left(\mathrm{sin}^{−\mathrm{1}} {f}\left({x}\right)\right). \\ $$$${if}\:{y}\left(\sqrt{\mathrm{3}}\right)=\frac{\Pi}{\mathrm{6}}\:{then}\:{y}\left(−\sqrt{\mathrm{3}}\right)=? \\ $$

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