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Question

If cos1x+cos1y+cos1z=π then, prove that x2+y2+z2+2xyz=1

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Solution

We have,

cos1x+cos1y+cos1z=π.....(1)

cos1x+cos1y=πcos1z

cos1[xy(1x2)(1y2)]=πcos1z

cos1A+cos1B=cos1[AB(1A2)(1B2)]

[xy(1x2)(1y2)]=cos(πcos1z)

xy(1x2)(1y2)=coscos1z

xy(1x2)(1y2)=z

xy+z=(1x2)(1y2)


On squaring both side and we get,

(xy+z)2=(1x2)(1y2)

x2y2+z2+2xyz=1x2y2+x2y2

x2+y2+z2+2xyz=1


Hence proved.


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