If xy-yz-zx-1- prove that x1-x2-y1-y2-z1-z2-4xyz1-x21-y21-z2-

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Trigonometric Ratios of Allied Angles

If xy+yz+zx...

Question

If $x y + y z + z x = 1$ , prove that $\frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{4 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})}$ .

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I f x y + y z + z x = 1,

\frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{4 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})} .

x y + y z + z x = 1

\frac{x}{1 + x^{2}} + \frac{y}{1 + y^{2}} + \frac{z}{1 + z^{2}} = \frac{2}{[(1 = x^{2}) (1 + y^{2}) (1 + z^{2})]^{1 / 2}}

1 :

x y + y z + z x = 1

x, y, z \in R^{+}

\frac{x}{1 + x^{2}} + \frac{y}{1 + y^{2}} + \frac{z}{1 + z^{2}} = \frac{2}{\sqrt{(1 + x^{2}) (1 + y^{2}) (1 + z^{2})}}

2 :

A B C

sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C

x^{2} + y^{2} + z^{2} - x y - y z - z x = \frac{1}{2} [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

\frac{(\frac{x}{y} - \frac{y}{x}) (\frac{y}{z} - \frac{z}{y}) (\frac{z}{x} - \frac{x}{z})}{(\frac{1}{x^{2}} - \frac{1}{y^{2}}) (\frac{1}{y^{2}} - \frac{1}{z^{2}}) (\frac{1}{z^{2}} - \frac{1}{x^{2}})}

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