I am attempting to solve the equation $\arccos(x)-\arcsin(x)=\arccos(\sqrt{3}x)$. Trying to use range of left side expression, since it is always decreasing, the range of the $\arccos(x)-\arcsin(x)$ is $\left[\pi/2-2\arcsin(1/\sqrt{3}),\pi/2+2\arcsin(1/\sqrt{3})\right]$. But that expression does not seem useful for comparison with the range of the expression $\arccos(\sqrt{3}x)$. I need to solve this without using the formula of $\arccos(x)-\arccos(y)$.
Any hints are appreciated. Thanks.
Best Answer
Hint: apply cos to both sides: $x\sqrt{1-x^2} + \sqrt{1-x^2}x = x\sqrt{3}$.