The number of solutions ofsin⁡$$sin$$−$$1$$⁡$$log1/2$$⁡$$x+2cos$$⁡$$sin$$−$$1$$⁡$$x2$$−$$32=0$$, is

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Infinity Learn · Accepted Answer

The two terms on the LHS of the given equationare meaningful, if−$$1$$≤$$log1/2$$⁡x≤$$1$$ and −$$1$$≤$$x2$$−$$32$$≤$$1$$⇒ $$2$$≥x≥$$12$$ and $$1$$≤x≤$$5$$⇒$$1$$≤x≤$$2Now$$,$$1$$≤x≤$$2$$⇒$$12$$≤$$x2$$≤$$1$$⇒−$$12$$≤$$x2$$−$$1$$≤$$0$$⇒−π$$6$$≤$$sin$$−$$1$$⁡$$x2$$−$$1$$≤$$0$$⇒$$32$$≤$$cos$$⁡$$sin$$−$$1$$⁡$$x2$$−$$1$$≤$$1$$⇒$$3$$≤$$2cos$$⁡$$sin$$−$$1$$⁡$$x2$$−$$1$$≤$$2$$⇒$$3$$≤$$2cos$$⁡$$sin$$−$$1$$⁡$$x2$$−$$1$$≤$$2Again$$,$$1$$≤x≤$$2$$⇒−$$1$$≤$$log1/2$$⁡x≤$$0$$⇒−π$$2$$≤$$sin$$−$$1$$⁡$$log1/2$$⁡x≤$$0$$⇒−$$1$$≤$$sin$$⁡$$sin$$−$$1$$⁡$$log1/2$$⁡x≤$$0$$⇒$$3$$−$$1$$≤$$sin$$⁡$$sin$$−$$1$$⁡$$log1/2$$⁡$$x+$$$$cos$$⁡$$sin$$−$$1$$⁡$$x2$$−$$32$$≤$$2$$⇒ $$sin$$⁡$$sin$$−$$1$$⁡$$log1/2$$⁡$$x+2cos$$⁡$$sin$$−$$1$$⁡$$x2$$−$$32$$≠$$0$$ for any xHence, the given equation has no solution.

The number of solutions of
$\sin \{\sin^{- 1} (\log_{1 / 2} x)\} + 2 |\cos \{\sin^{- 1} (\frac{x}{2} - \frac{3}{2})\}| = 0$ , is

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The number of solutions ofsin⁡sin−1⁡log1/2⁡x+2cos⁡sin−1⁡x2−32=0, is

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The number of solutions of
$\sin \{\sin^{- 1} (\log_{1 / 2} x)\} + 2 |\cos \{\sin^{- 1} (\frac{x}{2} - \frac{3}{2})\}| = 0$ , is