Can I apply Chinese remainder theorem here?

Distance between incentre and orthocentre. Find $\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$ Are there theoretical applications of trigonometry? Showing $A+B$ is invertible? Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $ Identity of a ring as two different sums of idempotents $x^2+y^2=2z^2$, positive integer solutions What is the current state of formalized mathematics? Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation? Does $AB+BA=0$ imply $AB=BA=0$ when $A$ is real semidefinite matrix? Why does it have to be an integer? Proof: Sum of dimension of orthogonal complement and vector subspace