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  • If sin^-1x+sin^-1y+sin^-1 z=pi , Prove that x(1-x^2)^1/2+y(1-y^2)^1/2+z(1-z^2)^1/2

If sin^-1x+sin^-1y+sin^-1 z=pi , Prove that x(1-x^2)^1/2+y(1-y^2)^1/2+z(1-z^2)^1/2

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Solution:  Let  sin^-1x=A ,  sin^-1y=B ,  sin^-1z=C

             Then x=sin A , y=sin B , z=sin C

       Now,  

             sin^-1x+sin^-1y+sin^-1z=pi

      Rightarrow     A+B+C=pi

     Rightarrow     sin 2A +sin 2B+sin 2C=4sin Acdot sin Bcdot sin C

     Rightarrow   2sin Acdot cos A+2sin Bcdot cos B+2sin Ccdotcos C=4sin A cdot sin B cdot sin C

     Rightarrow     xsqrt1-x^2+ysqrt1-x^2+zsqrt1-z^2=2xyz

Posted by

Deependra Verma

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