Question
Write the value of $\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\cos180^\circ.$
✓
Answer
Given that: $\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\cos180^\circ$
$=\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\cos180^\circ$
$=\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos89^\circ\cos90^\circ\cos91^\circ\dots\cos179^\circ$
$=\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos89^\circ\times0\dots\cos179^\circ\cos180^\circ$
$=0$ $\big[\cos90^\circ=0\big]$
Hence the value of $\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\cos180^\circ\text{ is }0$
$=\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\cos180^\circ$
$=\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos89^\circ\cos90^\circ\cos91^\circ\dots\cos179^\circ$
$=\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos89^\circ\times0\dots\cos179^\circ\cos180^\circ$
$=0$ $\big[\cos90^\circ=0\big]$
Hence the value of $\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\cos180^\circ\text{ is }0$
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