Cos9x-cos7x+cos3x-cosx=0

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вопрос

To solve this trigonometric equation, we can use the cosine addition and subtraction identities:

cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)
cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)

Given equation:
cos(9x) - cos(7x) + cos(3x) - cos(x) = 0

Apply the cosine subtraction identity to simplify the equation:
-2sin((9x + 7x)/2)sin((9x - 7x)/2) + 2sin((3x + x)/2)sin((3x - x)/2) = 0
-2sin(8x)sin(x) + 2sin(2x)sin(x) = 0

Factor out sin(x):
-2sin(x)[sin(8x) - sin(2x)] = 0

Set each factor equal to zero to find the solutions:
sin(x) = 0
This gives x = 0, π, 2π, ...

sin(8x) - sin(2x) = 0
sin(8x) = sin(2x)

Since sin(A) = sin(B) when A = nπ + (-1)^n*B (n is an integer), we solve for x:
8x = 2x + n(2π)
6x = n(2π)
x = n(π/3)

Therefore, the solutions to the equation cos(9x) - cos(7x) + cos(3x) - cos(x) = 0 are:
x = 0, π, 2π, ..., n(π/3) where n is an integer.