Is squaring of both sides of this equation allowed?

Question

Are we allowed to square both sides of this equation? ($x>0$) $$x^2 \sqrt{1-x^2} \sin (x)=\left(x^2+2\right) \cos (x)-x \sin (x)$$

Answer 1

There is a general rule for that.

Let $x, y \in \mathbb{R}$. Then $x = y \implies x^2 = y^2$.

Note that the converse doesn’t apply. For example, consider the number $25$. We have that $(-5)^2 = 25 = 5^2$ and $-5 \neq 5$.

Although note that you will get an extraneous solution, so at the end you must check what the solutions really are.