If x=asecθcosϕy=bsecθsinϕandz=ctanθ, show that x^2/a^2 + y^2/b^2− z^2/c^2 =1

If cos^(-1) x + cos^(-1) y + cos ^(-1) z = pi and 0 lt x,y,z lt 1 show that x^(2) +y^(2) +z^(2) +2xyz = 1

Let a ,b , c be the real numbers. The following system of equations in x ,y ,a n dz (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(a^2)=1,(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(a^2)=1,-(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(a^2)=1 has a. no solution b. unique solution c. infinitely many solutions d. finitely many solutions

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)

Show that |(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|=0.

If y= (tan^(-1)x)^(2) , show that (x^(2)+1)^(2) y_(2)+2x(x^(2)+1)y_(1)=2 .

If x+y+z=1 and x ,y ,z are positive, then show that (x+1/x)^2+(y+1/y)^2+(z+1/z)^2>(100)/3

IF sin^(-1) X+ sin^(-1) y+ sin^(-1) z= pi , then prove that x^4 +y^4+ z^4 +4x^2 y^2 z^2 = 2(x^2 y^2 +y^2 z^2+z^2x^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