Question

Solve $\tan 3x=\tan 5x$.

Answer 1

We re-write the equation as
$\frac{\sin 3x}{\cos 3x}-\frac{\sin 5x}{\cos 5x}=0$
or $\frac{\sin 3x\cos 5x-\sin 5x\cos 3x}{\cos 3x\cos 5x}=0$
or $\frac{\sin 2x}{\cos 3x\cos 5x}=0$.
Now solving the equation $\sin 2x=0$, we get $x=\frac{1}{2}n\pi ,n\in I$. But we must discard extraneous solutions, that is, those for which the denominator $\cos 3x\cos 5x$ vanishes, which clearly happens when n is odd. Thus the solution of the given equation will be given by $x=\frac{1}{2}n\pi$, where n is even, say $n=2m$, $m\in I$
Hence the required solution is $x=m\pi ,m\in I$
Quite obviously, it is a serious error to take the solution as $x=\frac{1}{2}n\pi ,n\in I$.