UPSC NDA I 2024 Mathematics Paper

$\cos^{-1} x = \sin^{-1} x$We know that the ranges of the inverse trigonometric functions limit the values of $x$. Specifically, for both $\cos^{-1} x$ and $\sin^{-1} x, x$ must be in the range $[-1,1]$.Let's denote the common value of $\cos^{-1} x$ and $\sin^{-1} x$ by $\theta$. Therefore, we have:$\theta = \cos^{-1} x = \sin^{-1} x$From the properties of inverse trigonometric functions, we know:$\cos \theta = x$$\sin \theta = x$We also know from trigonometric identities that:$\cos^2 \theta + \sin^2 \theta = 1$Substituting $x$ into the identity, we get:$x^2 + x^2 = 1$$2 x^2 = 1$$x^2 = \frac{1}{2}$$x = \pm \frac{1}{\sqrt{2}}$Since $x$ must fall within the range $[-1,1]$, both positive and negative values are valid within this context. However, given the problem's options, the correct answer aligns with only the positive value provided in the options list.