Question

$\mathrm{If}A+B+C=\pi \mathrm{and}{\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=a+\mathrm{bcosA}.\mathrm{cosB}.\mathrm{cosC},\mathrm{then}ab\mathrm{is}\mathrm{equal}\mathrm{to}$
.

• A. 1
• B. -1
• C. 2
• D. -2

Answer 1

The correct option is D -2

These types of questions are solved using conditional identities.
Here, ${\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=\frac{1+\cos 2A}{2}+\frac{1+\cos 2B}{2}+\frac{1+\cos 2C}{2}=\frac{1}{2}\left(\cos 2A+\cos 2B+\cos 2C\right)+\frac{3}{2}\mathrm{Using}\mathrm{the}\mathrm{conditional}\mathrm{identity},\mathrm{when}A+B+C=\pi \Rightarrow \cos 2A+\cos 2B+\cos 2C=-1-4\cos A.\cos B.\cos C\mathrm{we}\mathrm{get},=\frac{1}{2}(-1-4\cos A.\cos B.\cos C)+\frac{3}{2}=1-2\cos A.\cos B.\cos C$
Now, comparing with $a+b\cos A\cos B\cos C$
we get,
$a=1,b=-2\Rightarrow ab=-2.$