Question

If tanx + tany + tanz = Pi or Pi /2 then prove that... x+y+z=xyz






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Answer 1

We have to prove that, x + y + z = xyz or, xy + yz + zx = 1 when tan¯1x + tan¯¹y + tan¯¹z = π or, π/2

Proof : we know, tan(A + B + C) = (tanA + tanB + tanC - tanA tanB tanC)/(1 - tanA tanB - tanB tanC - tanC tanA)

so, tan¯¹x + tan¯¹y + tan¯¹z = tan¯¹[(x+ y + z - xyz)/(1 - xy - yz - zx)

If tan¯¹x + tan¯¹y + tan¯¹z = π

then, tan¯¹[(x + y + z - xyz)/(1 - xy - yz - zx)] = π

⇒(x + y + z - xyz)/(1 - xy - yz - zx) = tanπ = 0

⇒x + y + z = xyz [ hence proved]

Again, if tan¯¹x + tan¯¹y + tan¯¹z = π/2

⇒tan¯¹[(x + y + z - xyz)/(1 - xy - yz - zx)] = π/2

⇒(x + y + z - xyz)/(1 - xy - yz - zx) = tanπ/2 = 1/0

⇒1 - xy - yz - zx = 0

⇒xy + yz + zx = 1 [ hence proved ]