Sinx*sin(60-x)*sin(60+x)=1\8

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To solve this equation, we can use the trigonometric identity: sin(a)sin(b) = (1/2)[cos(a-b) - cos(a+b)].

Applying this identity to the given equation, we get:

sin(x)sin(60-x) = (1/2)[cos(x-(60-x)) - cos(x+(60-x))]
sin(x)sin(60-x) = (1/2)[cos(2x-60) - cos(120)]

Next, we use the double angle formula for cosine: cos(2x) = 1 - 2sin^2(x). Substituting this into the equation, we get:

sin(x)[1 - 2sin^2(x)] = (1/2)[1 - 2sin^2(x) - cos(120)]

Expanding and simplifying the equation gives:

sin(x) - 2sin^3(x) = (1/2)[1 - 2sin^2(x) + cos(120)]

Since cos(120) = -1/2, the equation becomes:

sin(x) - 2sin^3(x) = (1/2)[1 - 2sin^2(x) - 1/2]

Simplifying further, we get:

sin(x) - 2sin^3(x) = 2 - 4sin^2(x) - 1
sin(x) - 2sin^3(x) = 1 - 4sin^2(x)

Rearranging terms gives:

2sin^3(x) - 4sin^2(x) + sin(x) - 1 = 0

By factoring out sin(x) - 1, we get:

(sin(x) - 1)(2sin^2(x) + 2sin(x) + 1) = 0

This equation has one solution: sin(x) = 1.

Therefore, the solution to the given equation is sin(x) = 1.