Question

Tanxsecx-2tanx=0,for all real values of x​

Answer 1

We can factor $\tan(x)$ on the left hand side to get

$\tan(x)(\sec(x)-2)=0$

A factored equation equals zero if and only if at least one of the factor equals zero. So, this equation equals zero if and only if

$\tan(x)=0$

or

$\sec(x)-2=0$

In the first case, we have

$\tan(x)=0\iff x=k\pi$

In the second case, we have

$\sec(x)-2=0 \iff \dfrac{1}{\cos(x)}=2 \iff \cos(x)=\dfrac{1}{2}$

And so, the solutions are

$x=\pm\dfrac{\pi}{3}+2k\pi$