Question

if cos-1x + cos-1y + cos-1z = pi

prove that x2 + y2 + z2 +2xyz = 1

Answer 1

Let cos-1x =A then x= cosA

Let cos-1y=B then y =cosB

Let cos-1z=C then z=cosC

Then A+B+C =p

Xv(1-X2) +Yv(1-Y2) +Zv(1-Z2)=cosA v(1-cos2A)+cosBv(1-cos2B) +cosCv(1-cos2C)

=CosA.sinA+cosB.sinB+cosC.sinC

=1/2[cos2A+cos2B+cos2C]

=1/2 [4cosA.cosB.cosC]

=2cosA.cosB.cosC

=2xyz