Question

If $A+B+C={180}^o$, then
${\cos }^2\frac{A}{2}+{\cos }^2\frac{B}{2}+{\cos }^2\frac{C}{2}=2(1+\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2})$

• A. True
• B. False

Answer 1

The correct option is A True

Taking LHS

$co{s}^2\frac{A}{2}+co{s}^2\frac{B}{2}+co{s}^2\frac{C}{2}$

$co{s}^2\frac{A}{2}+1-si{n}^2\frac{B}{2}+co{s}^2\frac{C}{2}$

=$cos(\frac{A}{2}+\frac{B}{2}).cos(\frac{A}{2}-\frac{B}{2})+co{s}^2(\frac{A}{2}+\frac{B}{2})+1$

=$cos(\frac{A}{2}+\frac{B}{2})\left[cos\right(\frac{A}{2}-\frac{B}{2})-cos(\frac{A}{2}+\frac{B}{2}\left)\right]+1$

=$2sin\left(\frac{C}{2}\right)sin\left(\frac{A}{2}\right)sin\left(\frac{B}{2}\right)+2$

=$2[1+sin(\frac{C}{2}\left)sin\right(\frac{A}{2}\left)sin\right(\frac{B}{2}\left)\right]$