Question

In ∆ABC, if a², b² and c² are in A.P., prove that cot A, cot B and cot C are also in A.P. 

Answer 1

$$Let\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=k$$ 

Then, 

$$\sin A=ka,\sin B=kb,\sin C=kc$$ 

 a², b² and c² are in A.P. 

$$\Rightarrow 2b^2=a^2+c^2$$
$$\Rightarrow 2(a^2+c^2-b^2)=2(2b^2-b^2)=2b^2=b^2+b^2+c^2-a^2-c^2+a^2$$
$$\Rightarrow 2(a^2+c^2-b^2)=b^2+c^2-a^2+a^2+b^2-c^2$$
$$\Rightarrow \frac{2(a^2+c^2-b^2)}{2abc}=\frac{(b^2+c^2-a^2)}{2abc}+\frac{(a^2+b^2-c^2)}{2abc}$$
$$\Rightarrow \frac{2\cos B}{kb}=\frac{\cos A}{ka}+\frac{\cos C}{kc}$$
$$\Rightarrow \frac{2\cos B}{\sin B}=\frac{\cos A}{\sin A}+\frac{\cos C}{\sin C}$$
$$\Rightarrow 2\cot B=\cot A+\cot C$$
$$\Rightarrow \cot A,\cot Band\cot CareinAP.$$