If sin^-1 x + sin ^-1 y +sin^-1 z =3pi/2 then value of Sigma (x^101 + y^101 ) (x^505 + y ^ 505 ) /(x^303 + y^303 ) (x^606 +y^606)

Solution:    We have ,

$\sin^{-1}x+\sin^{-1 }y+\sin^{-1}z=\frac{3\pi}{2}$

It is possible only  When

$\Rightarrow$    $\sin^{-1}x=\sin^{-1}y=\sin^{-1}z=\frac{\pi}{2}.$

$\Rightarrow$           $x=y=z=1.$

Now ,  $\sum \frac{(x^{101}+y^{101})(x^{505}+y^{505})}{(x^{303}+y^{303})(x^{606}+y^{606})}=\sum \frac{(1+1)(1+1)}{(1+1)(1+1)}=\sum 1=(1+1+1)=3.$

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