Prove that `[(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(x+y+z)(x ,y ,z >0)`

Prove that |x^2x^2-(y-z)^2y z y^2y^2-(z-x)^2z x z^2z^2-(x-y)^2x y|=(x-y)(y-z)(z-x)(x+y+z)(x^2+y^2+z^2)dot

Verify that x ^(3) + y ^(3) + z ^(3) - 3xyz =1/2 (x + y + z) [(x-y)^(2) + (y-z) ^(2) + (z-x) ^(2) ]

If cos^(-1)x+cos^(-1)y+cos^(-1)z=pi,t h e n (a) x^2+y^2+z^2+x y z=0 (b) x^2+y^2+z^2+2x y z=0 (c) x^2+y^2+z^2+x y z=1 (d) x^2+y^2+z^2+2x y z=1

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

For 0 < varphigeqpi/2,ifx=sum_(n=0)^oocos^(2n)varphi,y=sum_(n=0)^oosin^(2n)varphi,t h e n (a) x y z=x z+y (b) x y z=x y+z (c) x y z=x+y+z (d) x y z=y z+x

If x/(tan (theta + alpha)) = y/(tan (theta + beta)) = z/tan(theta + gamma), "prove that" (x+y)/(x-y)sin^(2) (alpha - beta) + (y + z)/(y - z) sin^(2) (beta - gamma) + (z + x)/(z-x) sin^(2) (gamma - alpha ) = 0

By using properties of determinants , show that : {:|( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) |:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

The three adjacent surfaces of cuboid are x sq. unit, y sq. unit and z sq. unit. The length of its one of the diagonal is Option 1 (xy/z+yz/x+zx/y)^(1/2) Option 2 ( z x/ y + x y/ z + y z/ x )^(1/2) Option 3 ( x y/ z + y z/ x + z x/ y ) Option 4 ( z y/ x + y z/ x + x y/ z )

If x+y+z=x y z prove that (2x)/(1-x^2)+(2y)/(1-y^2)+(2z)/(1-z^2)=(2x)/(1-x^2)(2y)/(1-y^2)(2z)/(1-z^2)dot

The determinant |[x p+y, x, y], [y p+z, y, z],[0,x p+y, y p+z]|=0 if (a) x y z are in AP (b) x y z are in GP (c) x y z are in HP (d) xy yz zx are in AP