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=2z21+z2,y=2x21+x2,z=2y21+y2
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Solution
x=2z21+z2,y=2x21+x2,z=2y21+y2 Let x=tanα,y=tanβ,z=tanγ ⇒ tanα=2sin2γ;tanβ=2sin2α;tanγ=2sin2β ⇒8sinαsinβsinγ.cosαcosβcosγ=1 or sinαsinβsinγ=0 (multiplying ) ⇒sin2α.sin2β.sin2γ=1 or (sinαsinβsinγ)=0 so, x,y,z∈[{0,0,0}or{1,1,1}] (sin2α≤1)