Find the number of real solutions of the system of equations x-2z21 -z2- y-2x21 -x2- z-2y21 -y2

x=2z^21 +z^2, y=2x^21 +x^2, z=2y^21 +y^2 Let x = tan α , y = tan β , z = tan γ ⇒ tan α=2 sin^2 γ ; tan β= 2 sin^2 α ; tan γ=2 sin^2 β ⇒ ...

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Sum of Trigonometric Ratios in Terms of Their Product

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Question

Find the number of real solutions of the system of equations $x = \frac{2 z^{2}}{1 + z^{2}}, y = \frac{2 x^{2}}{1 + x^{2}}, z = \frac{2 y^{2}}{1 + y^{2}}$

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\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1, \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1, - \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

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