Substitute for evaluation of determinant of the matrix `[[sinx,sin2x,sin3x],[siny,sin2y,sin3y],[sinz,sin2z,sin3z]]`

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You need to evaluate the determinant of the given matrix:

Delta = `[[sinx,sin2x,sin3x],[siny,sin2y,sin3y],[sinz,sin2z,sin3z]]`

Use the following formulas: `sin 2alpha = 2 sin alpha*cos alpha`

`sin 3 alpha = 3 sin alpha - 4 sin^3 alpha`

Write the new form of the matrix:

`Delta = [[sin x, 2sin x*cos x, 3 sinx - 4sin^3 x],[sin y, 2sin y*cos y, 3 siny - 4sin^3 y], [sin z, 2sin z*cos z, 3 sin z- 4sin^3 z]]`

Notice the common factors: sin x*sin y*sin z

`Delta = 2sin x*sin y*sin z*[[1, cos x, 3 - 4sin^2 x],[1, cos y, 3 - 4sin^2 y], [1, cos z, 3 - 4sin^2 z]]`

Use the basic formula of trigonometry: `sin^2 x = 1 - cos^2 x`  such that:

`Delta = 2sin x*sin y*sin z*[[1, cos x, 3 - 4 + 4cos^2 x],[1, cos y, 3 - 4 + 4cos^2 y], [1, cos z, 3 - 4 + 4cos^2 z]]`

`Delta = 2sin x*sin y*sin z*[[1, cos x,-1 + 4cos^2 x],[1, cos y,-1 + 4cos^2 y], [1, cos z, -1 + 4cos^2 z]]`

Adding the first column to the last column yields:

`Delta = 2sin x*sin y*sin z*[[1, cos x, 4cos^2 x],[1, cos y,4cos^2 y], [1, cos z, 4cos^2 z]]`

`Delta = 4*2sin x*sin y*sin z*[[1, cos x, cos^2 x],[1, cos y,cos^2 y], [1, cos z, cos^2 z]]`

Evaluating the Vandermonde's determinant yields:

`Delta`  = 8sin x*sin y*sin z*(cos y - cos x)(cos z - cos x)(cos z - cos y)

Evaluating the determinant yields: `Delta`  = `8sin x*sin y*sin z*(cos y - cos x)(cos z - cos x)(cos z - cos y)`

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