" 7- prove that "|[1,x,x^(2)],[x^(2),1,x],[x,x^(2),1]|=(1-x^(3))^(2)

Prove det[[1,x,x^(2)x^(2),1,xx,x^(2),1]]=(1-x^(3))^(2)

Prove that |(1,x,x^2),(x^2,1,x),(x,x^2,1)|=(1-x^2) .

Prove the following : [[1,x,x^2],[x^2,1,x],[x,x^2,1]]=(1-x^3)^2

Prove that: |[1+x_(1),x_(2),x_(3)],[x_(1),1+x_(2),x_(3)],[x_(1),x_(2),1+x_(3)]|=1+x_(1)+x_(2)+x_(3)

prove that |(1,x,x+1),(2x,x(x-1),x(x+1)),(3x(1-x),x(x-1)(x-2),x(x+1)(x-1))|=6x^(2)(1-x^(2))

For x>0 Prove that x-(x^(2))/(2)

Using the properties of determinants, prove that following : |{:(1,x,x+1),(2x,x(x-1),x(x+1)),(3x(1-x),x(x-1)(x-2),x(x+1)(x-1)):}|=6x^(2)(1-x^(2))

Prove that tan^(-1)x+"tan"^(-1)(2x)/(1-x^(2))=tan^(-1)((3x-x^(3))/(1-3x^(2))),|x|lt1/(sqrt(3))

Prove that tan^(-1)x+"tan"^(-1)(2x)/(1-x^(2))=tan^(-1)((3x-x^(3))/(1-3x^(2))),|x|lt1/(sqrt(3))