Sine vs Cosine

The arc length $L$ of a curve $y=f(x)$ between $x=a$ and $x=b$ is given by $L=\int_a^b\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}\; dx.$ Thus, we seek the maximum possible value of $\dfrac{\int_0^t\sqrt{1+\sin^2(x)}\; dx}{\int_0^t\sqrt{1+\cos^2(x)}\; dx},$ for some value of $t$ . By inspecting the graphs of $y=\sqrt{1+\cos^2(x)}$ and $y=\sqrt{1+\sin^2(x)}$ , it is easy to see that the maximum must occur when $\tfrac{\pi }{4}<t<\tfrac{3\pi }{4}$ , but other than this, I found no way to evaluate the derivative of this ratio analytically (anyone else find a way?)

Solving numerically gives $t\approx 2.2449126574055164726$ and $R\approx 1.0764947166963004306$ , meaning that $\lfloor 1000R\rfloor =\boxed{1076}$ .