If $8\sin x - \cos x=4$, then find possible values of $x$ I am not understanding what exactly can watch do here. First I thought that if I could square it but it was in vain. Please help me.

Put $t = \tan \frac{x}{2}$. Then $\sin x = \frac{2t}{1+t^2}, \cos x = \frac{1-t^2}{1+t^2}$. The equation reduces to
$$3t^2 - 16t + 5 = 0$$ and hence $(3t-1)(t-5) = 0$. Hence the solutions are given by $\tan \frac{x}{2} = 5$ or $\frac{1}{3}$
Let us verify the solutions graphically. The following shows the graphs of $8 \sin(x) - \cos(x) = 5$ (green curve), $\tan \frac{x}{2} = 5$ (blue curve) and $\tan \frac{x}{2} = \frac{1}{3}$ (red curve). It is clear that we have obtained all the solutions.

