^ - 1 \ (1)/(2)x^ 2 + 1 - x^ 2 1 - x^ 2 4 \ = ^ - 1 (x)/(2) - ^ - 1...

It is given that& $\cos^{- 1}\left(\left\{\dfrac{x2}{2} + \sqrt{1 - x^{2}}\sqrt{1 - \dfrac{x2}{4}}\right\}\right) = \cos^{- 1} \left(\dfrac{x}{2}\right) - \cos^{- 1}\left(x\right)$

Since, $0$\le \cos^{- 1}$\left(\dfrac{x^{2}}{2} + \sqrt{1 - x^{2}}\sqrt{1 - \dfrac{x^{2}}{4}}\right) \le \dfrac{\pi}{2}<br /><br />Because$\cos^{- 1} x$is in first quadrant when$x$is positive<br /><br />And$\cos^{- 1} $\dfrac{x}{2} - \cos^{- 1}$ x ≥ 0$<br /><br />So,$\cos^{- 1} $\dfrac{x}{2} \ge \cos^{- 1}$ x$<br /><br />Also, \left|\dfrac{x}{2}\right| \le$1⇒ | x | ≤ 1$</p>$