Question

If $x+y+z=xyz$, prove that $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{rx}{1-x^2}\frac{py}{1-y^2}\frac{qz}{1-z^2}$ where $p,r,q$ are constants. Find value of $pqr$

Answer 1

Let $x=\tan A$, $y=\tan B$ and $z=\tan C$
Now $x+y+z=xyz$
$\Rightarrow$ $\tan A+\tan B+\tan C=\tan A\tan B\tan C$
$\Rightarrow$ $A+B+C=n\pi$
or $2A+2B+2C=2n\pi$
or $\tan 2A+\tan 2B+\tan 2C=\tan 2A\tan 2B\tan 2C$
or $=\frac{2\tan A}{1-{\tan }^{2}A}+\frac{2\tan B}{1-{\tan }^{2}B}+\frac{2\tan C}{1-{\tan }^{2}C}=\frac{2\tan A}{1-{\tan }^{2}A}\frac{2\tan B}{1-{\tan }^{2}B}\frac{2\tan C}{1-{\tan }^{2}C}$
$\Rightarrow$ $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\frac{2y}{1-y^2}\frac{2z}{1-z^2}$
Ans: 8