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If $A+B+C=60^{\circ}$, then
$\cos \left(30^{\circ}-A\right)+\cos \left(30^{\circ}-B\right)+\cos \left(30^{\circ}-C\right)$
$+\sin (A+B+C)=$
Options:
$\cos \left(30^{\circ}-A\right)+\cos \left(30^{\circ}-B\right)+\cos \left(30^{\circ}-C\right)$
$+\sin (A+B+C)=$
Solution:
1725 Upvotes
Verified Answer
The correct answer is:
$4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
For $A+B+C=60^{\circ}$, then
$\left(\cos 30^{\circ}-A\right)+\cos \left(30^{\circ}-B\right)+\cos \left(30^{\circ}-C\right)$ $+\sin (A+B+C)$
$=2 \cos \left(30^{\circ}-\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)+\cos \left(30^{\circ}-C\right)$ $+\cos 30^{\circ}$
$=2 \cos \frac{C}{2} \cos \frac{A-B}{2}+2 \cos \left(30^{\circ}-\frac{C}{2}\right) \cos \left(\frac{C}{2}\right)$
$=2 \cos \frac{C}{2}\left[\cos \frac{A-B}{2}+\cos \left(30^{\circ}-\frac{C}{2}\right)\right]$
$=2 \cos \frac{C}{2}$ $\left[2 \cos \left(\frac{A-B-C}{4}+15^{\circ}\right) \cos \left(\frac{A-B+C}{4}-15^{\circ}\right)\right]$
$=4 \cos \frac{C}{2} \cos \left(\frac{A+60^{\circ}-B-C}{4}\right) \cos \left(\frac{A+C-60^{\circ}-B}{4}\right)$
$=4 \cos \frac{C}{2} \cos \left(\frac{2 A}{4}\right) \cos \left(\frac{-2 B}{4}\right)=4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
$\left(\cos 30^{\circ}-A\right)+\cos \left(30^{\circ}-B\right)+\cos \left(30^{\circ}-C\right)$ $+\sin (A+B+C)$
$=2 \cos \left(30^{\circ}-\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)+\cos \left(30^{\circ}-C\right)$ $+\cos 30^{\circ}$
$=2 \cos \frac{C}{2} \cos \frac{A-B}{2}+2 \cos \left(30^{\circ}-\frac{C}{2}\right) \cos \left(\frac{C}{2}\right)$
$=2 \cos \frac{C}{2}\left[\cos \frac{A-B}{2}+\cos \left(30^{\circ}-\frac{C}{2}\right)\right]$
$=2 \cos \frac{C}{2}$ $\left[2 \cos \left(\frac{A-B-C}{4}+15^{\circ}\right) \cos \left(\frac{A-B+C}{4}-15^{\circ}\right)\right]$
$=4 \cos \frac{C}{2} \cos \left(\frac{A+60^{\circ}-B-C}{4}\right) \cos \left(\frac{A+C-60^{\circ}-B}{4}\right)$
$=4 \cos \frac{C}{2} \cos \left(\frac{2 A}{4}\right) \cos \left(\frac{-2 B}{4}\right)=4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
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