If $$\sin^{-1}\Big(x-\frac{x^2}{2} +\frac{x^4}{4} ...\infty \Big)+\cos^{-1}\Big(x^2 -\frac{x^4}{2} +\frac{x^6}{4} ...\infty \Big) = \frac{\pi}{2}$$ where $0<|x|<1.414$ then $x$ equals:
Now I tried to use $\sin^{-1}(x) + \cos^{-1}(x) =\frac{\pi}{2}$ and thus equated the equations inside the inverse trigonometric functions in the questions. But neither am I getting anything from this nor could I find any other method. Please help. Cheers!!
Notice that for the equation $\sin^{-1}(a)+\cos^{-1}(b) = \pi/2$, we can take the sin of both sides, coupled with the identity $\sin(a+b) = \sin(a)\cos(b)+\cos(a)\sin(b)$ to find that $$a - a^2 + b - b^2 = -1$$. If you can find closed-form expressions for a and b for your problem, similar to Jacky's comment, you could try solving it this way.