Find angle between $y=\sin x$ and $y=\cos x$ at their intersection point. Find angle between $y=\sin x$ and $y=\cos x$ at their intersection point.
Intersection points are $\frac{\pi}{4}+\pi k$ and to find angle between them we need to compute derivatives at intersection points but then I can't combine them to get an answer which is $\arctan2\sqrt2$. Will be thankful for your help.

If you think about these as parametrized curves $(t,\sin t)$ and $(t,\cos t)$, then the angle between them at a point $t_0$ is the angle between $(1,\cos t_0)$ and $(1,-\sin t_0)$. If $t_0 = \frac{\pi}{4} + \pi k$, then we get
$$\cos\theta = \frac{(1,\pm\frac{1}{\sqrt{2}})\cdot(1,\mp\frac{1}{\sqrt{2}})}{\frac{3}{2}} = \frac{1}{2}\cdot\frac{2}{3} = \frac{1}{3}.$$
This, of course, yields an angle of $\arccos\frac{1}{3}$, which is the same as $\arctan 2\sqrt{2}$.
