Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $\cos A+\cos B+2\cos C=2$ then the sides of the $\Delta ABC$ are in?

• A. AP
• B. GP
• C. HP
• D. none of these

Answer 1

The correct option is A AP

Given, $\cos A+2\cos B+\cos C=2\Rightarrow \cos A+\cos C=2(1-\cos B)$
$\Rightarrow 2\cos \left(\frac{A+C}{2}\right)\cos \left(\frac{A-C}{2}\right)=2\times 2{\sin }^2\frac{B}{2}$
$\Rightarrow 2\sin \frac{B}{2}\cos \left(\frac{A-C}{2}\right)=2\times 2\sin \frac{B}{2}\cos \left(\frac{A+C}{2}\right)$
$\Rightarrow \cos \left(\frac{A-C}{2}\right)=2\cos \left(\frac{A+C}{2}\right)$
$\Rightarrow \cos \frac{A}{2}\cos \frac{C}{2}=3\sin \frac{A}{2}\sin \frac{C}{2}$
$\Rightarrow \tan \frac{A}{2}\tan \frac{C}{2}=3$
$\Rightarrow \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\times \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}=\frac{1}{3}$
$\Rightarrow 3(s-b)=s\Rightarrow 2s=3b$
$\Rightarrow a+b+c=3b\Rightarrow b=\frac{a+c}{2}$
Hence $a,b,c$ are in A.P