Prove that: 5 4 9 4 + 17 4 15 4 =0 - Vidyadip

Question

Prove that: $\tan\frac{5\pi}{4}\cot\frac{9\pi}{4}+\tan\frac{17\pi}{4}\cot\frac{15\pi}{4}=0$

✓

Answer

$\text{L.H.S}=\tan\frac{5\pi}{4}\cot\frac{9\pi}{4}+\tan\frac{17\pi}{4}\cot\frac{15\pi}{4}$ $=\tan225^\circ\cot405^\circ+\tan765^\circ\cot675^\circ$ $=\tan\Big(\pi+\frac{\pi}{4}\Big)\cot\Big(2\pi+\frac{\pi}{4}\Big)+\tan\Big(4\pi+\frac{\pi}{4}\Big)\cot\Big(4\pi-\frac{\pi}{4}\Big)$ $=\tan\frac{\pi}{4}\cot\frac{\pi}{4}+\tan\frac{\pi}{4}\Big(-\cot\frac{\pi}{4}\Big)$ $=1.1+1.(-1)$ $=1-1$ $=0$ $=\text{R.H.S}$ $\text{Proved}$

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