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You need to know the trigonometric formulae for sin and cos of compound angles.

sin(x+y) = sin(x)*cos(y)+cos(x)*sin(y)

and

cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)

Using these, y = x implies that

sin(2x) = sin(x+x) = 2*sin(x)cos(x)

and

cos(2x) = cos(x+x) = cos^2(x) - sin^2(x)

Next, the triple angle formulae are:

sin(3x) = sin(2x + x) = 3*sin(x) - 4*sin^3(x)

and

cos(3x) = 4*cos^3(x) - 3*cos(x)

Then the left hand side

= 2*[3*sin(x) - 4*sin^3(x)]/sin(x) + 2*[4*cos^3(x) - 3*cos(x)]/cos(x)

= 6 - 8*sin^2(x) + 8cos^2(x) - 6

= 8*[cos^2(x) - sin^2(x)]

= 8*cos(2x) = right hand side.

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