The number of solutions of the equation sin-1x2+13+cos-1x2-23=x2, forx∈-1,1 and [x] denotes the greatest integer less than or equal to x, is:

x2-23=x2+13-1=x2+13-1 -1≤x2+13≤1 ⇒ x2+13=-1, 0, 1 x2+13=-1 ⇒not possible x2+13=0 ⇒0+π=x2 ⇒no solution x2+13=0 ⇒π2+π2=x2 ⇒ no solution

The number of solutions of the equation sin-1x2+13+cos-1x2-23=x2, forx∈-1,1 and [x] denotes the greatest integer less than or equal to x, is:

x2-23=x2+13-1=x2+13-1 -1≤x2+13≤1 ⇒ x2+13=-1, 0, 1 x2+13=-1 ⇒not possible x2+13=0 ⇒0+π=x2 ⇒no solution x2+13=0 ⇒π2+π2=x2 ⇒ no solution

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The number of solutions of the equation $\sin^{- 1} [x^{2} + \frac{1}{3}] + \cos^{- 1} [x^{2} - \frac{2}{3}] = x^{2},$ for
$x \in [- 1, 1]$ and [x] denotes the greatest integer less than or equal to x, is:

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Infinite

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Detailed Solution

$[x^{2} - \frac{2}{3}] = [x^{2} + \frac{1}{3} - 1] = [x^{2} + \frac{1}{3}] - 1 - 1 \leq [x^{2} + \frac{1}{3}] \leq 1 \Rightarrow [x^{2} + \frac{1}{3}] = - 1, 0, 1 [x^{2} + \frac{1}{3}] = - 1 \Rightarrow not possible [x^{2} + \frac{1}{3}] = 0 \Rightarrow 0 + π = x^{2} \Rightarrow no solution [x^{2} + \frac{1}{3}] = 0 \Rightarrow \frac{π}{2} + \frac{π}{2} = x^{2} \Rightarrow no solution$