Question

# If α^{2}, b^{2}, c^{2} are A.P then cot A, cot B, cot C are in^{}

- A.P
- G.P
- H.P
- A.G.P

Hint:

### The **law of sines** says that the ratio of the sine of one angle to the opposite side is the same ratio for all three angles.

## The correct answer is: A.P

### If α^{2}, b^{2}, c^{2} are A.P then cot A, cot B, cot C are in

α^{2}, b^{2}, c^{2} are A.P

It means a^{2} + c^{2} = 2b^{2} .

b^{2 }- a^{2} = c^{2} - b^{2}
$⇒k2sin2B−k2sin2A=k2sin2C−k2sin2B$ using sine rule $a=ksinA,b=ksinB$ and $c=ksinC$
$⇒sin(B+A)sin(B−A)=sin(C+B)sin(C−B)$
$⇒sin(π−C)sin(B−A)=sin(π−A)sin(C−B)$ since $A+B+C=π$
$⇒sinCsin(B−A)=sinAsin(C−B)$
$⇒sinAsin(B−A) =sinCsin(C−B) $
$⇒sinAsinBsin(B−A) =sinBsinCsin(C−B) $
$⇒sinAsinBsinBcosA−cosBsinA =sinBsinCsinCcosB−cosBsinC $
$⇒cotA−cotB=cotB−cotC$
$∴cotA,cotB,cotC$ are in A.P

^{2 }- a

^{2}= c

^{2}- b

^{2}

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