First slide

Arithmetic progression

Question

if, $a^{2}, b^{2}, c^{2}$ are in AP, then which of the following is also in AP ?

Moderate

A

$\sin A, \sin B, \sin C$
B

$\tan A, \tan B, \tan C$
C

$\cot A, \cot B, \cot C$
D

None of these

Solution

$∵ a^{2}, b^{2}, c^{2}$ are in AP.
$\begin{aligned} \Rightarrow \sin^{2} B - \sin^{2} A = \sin^{2} C - \sin^{2} B \\ \Rightarrow \sin C (\sin Bcos A - \cos Bsin A) \end{aligned}$
$= \sin A (\sin Ccos B - \cos Csin B)$
On dividing by sin $A$ sin B sin C, we get
$2 \cot B = \cot A + \cot C$
$\Rightarrow \cot A, \cot B, \cot C$ are in AP.

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