Prove the following identities \[ \left|\begin{array}{ccc} z & x & y \\ z^{2} & x^{2} & y^{2} \\...

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If \( \left|\begin{array}{ccc}x^{n} & x^{n+2} & x^{n+4} \\ y^{n} & y^{n+2} & y^{n+4} \\ z^{n} & .... If \( \left|\begin{array}{ccc}x^{n} & x^{n+2} & x^{n+4} \\ y^{n} & y^{n+2} & y^{n+4} \\ z^{n} & ....
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Let \(f(x)=\left|\begin{array}{ccc}\sec x & x^2 & x \\ 2 \sin x & x^3 & 2 x^2 \\... Let \(f(x)=\left|\begin{array}{ccc}\sec x & x^2 & x \\ 2 \sin x & x^3 & 2 x^2 \\...
If \( D=\left|\begin{array}{ccc}\frac{1}{z} & \frac{1}{z} & -\frac{(x+y)}{z^{2}} \\ -\frac{(y+z)... If \( D=\left|\begin{array}{ccc}\frac{1}{z} & \frac{1}{z} & -\frac{(x+y)}{z^{2}} \\ -\frac{(y+z)...