Prove that the value of each the following determinants is zero: `|[(a^x+a^(-x))^2,(a^x-a^(-x))^2 ,1],[(b^y+b^(-y))^2,(b^y-b^(-y))^2 ,1],[(c^z+c^(-z))^2,(c^z-c^(-z))^2, 1]|`

Prove that the value of each the following determinants is zero: |[a-b,b-c,c-a],[ x-y, y-z, z-x],[ p-q, q-r ,r-p]|

Show that abs(((a-x)^2,(a-y)^2,(a-z)^2) , ((b-x)^2,(b-y)^2,(b-z)^2) , ((c-x)^2,(c-y)^2, (c-z)^2)) = 2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)

If x/(y+z)=y/(z+x)=z/(x+y) prove that the value of each ratio is either 1/2 or -1.

Eliminate x,y and z from the following equations: (x)/(y+z)=a,y/(z+x)=b,z/(x+y)=c

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)/(c^2)=1,(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1,-(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1 has a. no solution b. unique solution c. infinitely many solutions d. finitely many solutions

If x=a sec theta cosphi, y = b sec theta sin phi and z=c tan theta , then prove that (x^(2))/(a^(2))+(y^(2))/(b^(2))-(z^(2))/(c^(2))=1.

Using the property of determinants and without expanding {:|( x,a,x+a),( y,b,y+b),(z,c,z+c)|:} =0

If x/ (y+z) = y/(z+x) =z/(x+y) , then prove that the value of each ratio is either 1/2 or -1

If a tan x+ b tan y+ c tan z=m , then show that the minimum value of tan^(2)x+tan^(2)y+tan^(2)z "is" (m^(2))/(a^(2)+b^(2)+c^(2)) .

If a,b,c are in A.P and x,y,z are in G.P then prove that a^((b-c)log_(a)^(x))xxb^((c-a)log_(b)^(y))xxc^((a-b)log_(c )^(z))=1