If a,b,c are positives such that $a+b+c=\frac{\pi}{2}$ and $\mathrm{cot}\left(a\right),\mathrm{cot}\left(b\right),\mathrm{cot}\left(c\right)$ is in arithmetic progression, find $\mathrm{cot}\left(a\right)\mathrm{cot}\left(c\right)$

Miguel Davenport
2022-01-24
Answered

If a,b,c are positives such that $a+b+c=\frac{\pi}{2}$ and $\mathrm{cot}\left(a\right),\mathrm{cot}\left(b\right),\mathrm{cot}\left(c\right)$ is in arithmetic progression, find $\mathrm{cot}\left(a\right)\mathrm{cot}\left(c\right)$

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Dakota Cunningham

Answered 2022-01-25
Author has **9** answers

As

$\mathrm{cot}(x+y)=\frac{\mathrm{cot}x\mathrm{cot}y-1}{\mathrm{cot}x+\mathrm{cot}y}$

$\mathrm{cot}(a+c)=\mathrm{cot}(\frac{\pi}{2}-b)=\mathrm{tan}b=\frac{1}{\mathrm{cot}b}=\frac{\mathrm{cot}a\mathrm{cot}c-1}{\mathrm{cot}a+\mathrm{cot}c}$

$\frac{1}{\mathrm{cot}b}=\frac{\mathrm{cot}a\mathrm{cot}c-1}{2\mathrm{cot}b}$

So,

$\mathrm{cot}a\mathrm{cot}c=1+2=3$

So,

sjkuzy5

Answered 2022-01-26
Author has **11** answers

Write A=2a etc.

$\mathrm{cot}\frac{A}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}=\frac{s(s-a)}{\mathrm{\u25b3}}$

We have$s-a+s-c=2(s-b)$

$a+c=2b\Rightarrow 2s=a+b+c=3b$

$\mathrm{cot}\frac{A}{2}\mathrm{cot}\frac{C}{2}=\frac{s}{s-b}=3\text{}\text{as}\text{}3b=2s$

We have

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