Question
Without using trigomentric tables, prove that:
$\cos54^\circ\cos36^\circ+\sin54^\circ\sin36^\circ=0$
$\cos54^\circ\cos36^\circ+\sin54^\circ\sin36^\circ=0$
✓
Answer
$\text{L.H.S}=\cos54^\circ\cos36^\circ+\sin54^\circ\sin36^\circ$
$=\cos\big(90^\circ-36^\circ\big)\cos36^\circ+\sin\big(90^\circ-36^\circ\big)\sin36^\circ$
$=\sin36^\circ\cos36^\circ+\cos36^\circ\sin36^\circ$
$=0$
$=\text{R.H.S.}$
$=\cos\big(90^\circ-36^\circ\big)\cos36^\circ+\sin\big(90^\circ-36^\circ\big)\sin36^\circ$
$=\sin36^\circ\cos36^\circ+\cos36^\circ\sin36^\circ$
$=0$
$=\text{R.H.S.}$
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