Prove that: (-225^ ) (-405^ )- (765^ )+ (675^ )=0 - Vidyadip

Question

Prove that:
$\tan(-225^\circ)\cot(-405^\circ)-\tan(765^\circ)+\cot(675^\circ)=0$

✓

Answer

$\text{L.H.S}=\tan(-225^\circ)\cot(-405^\circ)-\tan(-765)\cot(675^\circ)$
$=-\tan225^\circ(-\cot405^\circ)+\tan765^\circ\cot765\circ$ $\begin{pmatrix}\because\tan(-\theta)=-\tan\theta\\\&\cot(-\theta)=-\cot\theta\end{pmatrix}$
$=\tan\Big(\pi+\frac{\pi}{4}\Big)\cot\Big(2\pi\frac{\pi}{4}\Big)+\tan\Big(4\pi+\frac{\pi}{4}\Big)\cot\Big(4\pi-\frac{\pi}{4}\Big)$
$=\tan\frac{\pi}{4}\cot\frac{\pi}{4}+\tan\frac{\pi}{4}\times\Big(-\cot\frac{\pi}{4}\Big)$ $(\because\cot(4\pi-\theta)=-\cot\theta)$
$= 1. 1 + 1 (-1) $
$= 1 - 1$
$= 0$
$=\text{R.H.S}$
$\text{Proved}$

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