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If $a < \frac{1}{32},$ then the number of solutions of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3} = a π^{3},$ is

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Solution

We find that
$\begin{array}{l} {(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3} \\ = {(\sin^{- 1} x + \cos^{- 1} x)}^{3} - 3 \sin^{- 1} x \cos^{- 1} x (\sin^{- 1} x + \cos^{- 1} x) \end{array}$
$\begin{array}{l} = \frac{π^{3}}{8} - 3 (\sin^{- 1} x \cos^{- 1} x) \frac{π}{2} \\ = \frac{π^{3}}{8} - \frac{3 π}{2} (\frac{π}{2} - \sin^{- 1} x) \sin^{- 1} x \\ = \frac{π^{3}}{8} - \frac{3 π^{2}}{4} \sin^{- 1} x + \frac{3 π}{2} {(\sin^{- 1} x)}^{2} \end{array}$
$\begin{array}{l} = \frac{π^{3}}{8} + \frac{3 π}{2} \{{(\sin^{- 1} x)}^{2} - \frac{π}{2} \sin^{- 1} x\} \\ = \frac{π^{3}}{8} + \frac{3 π}{2} \{{(\sin^{- 1} x - \frac{π}{4})}^{2}\} - \frac{3 π^{3}}{32} \\ = \frac{π^{3}}{32} + \frac{3 π}{2} {(\sin^{- 1} x - \frac{π}{4})}^{2} \end{array}$
$\begin{array}{l} \Rightarrow {(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3} \geq \frac{π^{3}}{32} \\ \Rightarrow a π^{3} \geq \frac{π^{3}}{32} \Rightarrow a \geq \frac{1}{32} \end{array}$
But, it is given that $a < \frac{1}{32} .$
Hence, for $a < \frac{1}{32},$ the equation has no solution.

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Introduction to ITF

Remember concepts with our Masterclasses.

If a<132, then the number of solutions of sin−1⁡x3+cos−1⁡x3=aπ3, is

Ready to Test Your Skills?

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If $a < \frac{1}{32},$ then the number of solutions of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3} = a π^{3},$ is