An object moves along a line modeled by the x-axis. Its position (i.e. x-coordinate) after $t$ seconds is given by \begin{equation} x(t)=a\sin t + b \cos t \end{equation} where $a$ and $b$ are positive constants.
Looking at the information, I have a question in my mind: how to prove that the sum of the velocity and squares of the position is always a constant? I know that the velocity is $x'(t)=a \cos t - b \sin t$. I try to calculate the given value and use trig identities to see whether the value's a real number. Is my direction correct? If no, any other hints to prove that?
Maybe there is a typo: I think the problem refers to the sum of the squares of the position and speed.
If that is the case, the result is indeed true:
\begin{align*} x^{2}(t) + v^{2}(t) & = (a\sin(t) + b\cos(t))^{2} + (a\cos(t) - b\sin(t))^{2}\\\\ & = a^{2}(\sin^{2}(t) + \cos^{2}(t)) + b^{2}(\cos^{2}(t) + \sin^{2}(t)) = a^{2} + b^{2} \end{align*}
and we are done.
Hopefully this helps !