Find the value of

To find: Value of $\tan \frac{-25 \pi}{3}$

We know that,

$\tan (-\theta)=-\tan \theta$

$\therefore \tan \left(-\frac{25 \pi}{3}\right)=-\tan \left(\frac{25 \pi}{3}\right)$

$\tan \left(-\frac{25 \pi}{3}\right)=-\tan \left(\frac{25 \pi}{3}\right)=-\tan \left(8 \pi+\frac{1}{3} \pi\right)$

$=-\tan \left(4 \times(2 \pi)+\frac{1}{3} \pi\right)$

Value of $\tan \mathrm{x}$ repeats after an interval of $2 \pi$, hence ignoring $4 \times(2 \pi)$

$=-\tan \left(\frac{1}{3} \pi\right)$

$=-\tan \left(\frac{1}{3} \times 180^{\circ}\right)$

$=-\tan 60^{\circ}$

$=-\sqrt{3}$

${\left[\because \tan 60^{\circ}=\sqrt{3}\right] }$