Question

If $a^2,b^2,c^2$ are in A.P., then $\cot A,\cot B,\cot C$ are in

• A. A.P
• B. G.P
• C. H.P
• D. None of these

Answer 1

Answer: (A)

A.P

Given $a^2,b^2,c^2$ are in A.P.

$b^2-a^2=c^2-b^2$

$k^{2}si{n}^{2}B-k^{2}si{n}^{2}A$ = $k^{2}si{n}^{2}C-k^{2}si{n}^{2}B$

$sin(B+A)\times sin(B-A)=sin(C+B)\times sin(C-B)$

$sinC\times sin(B-A)=sinA\times sin(C-B)$

$\therefore \frac{sin(B-A)}{sinA}=\frac{sin(C-B)}{sinC}$

$\therefore \frac{sin(B-A)}{sinA\times sinB}=\frac{sin(C-B)}{sinC\times sinB}$

$\frac{sinB\times cosA-cosB\times sinA}{sinA\times sinB}=\frac{sinC\times cosB-cosC\times sinB}{sinC\times sinB}$

$\therefore cotA-cotB=cotB-cotC$

Thus $cotA,cotB,cotC$ are in A.P.