Cos 3x - cos 5x= sin 4x

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вопрос

To solve the given trigonometric equation, we can use the trigonometric identity:

Cos(a) - Cos(b) = 2Sin((a+b)/2)Sin((a-b)/2)

Therefore, the equation becomes:

2Sin(4x)/2 * Sin(-x)/2 = Sin(4x)

Which simplifies to:

Sin(2x) * Sin(2x) = Sin(4x)

Sin^2(2x) = Sin(4x)

An identity to be applied here is:

Sin^2(x) = (1 - Cos(2x)) / 2

Thus, the equation becomes:

(1 - Cos(4x)) / 2 = Sin(4x)

Solving for Cos(4x) yields:

Cos(4x) = 1 - 2Sin(4x)

An identity to be applied is:

Sin^2(x) + Cos^2(x) = 1

Therefore, the solution is Cos(4x) = 1 - 2(1 - Cos^2(2x))

Solution:

Cos(4x) = 1 - 2 + 2Cos^2(2x)
Cos(4x) = -1 + 2Cos^2(2x)

Hence, the solution to the given trigonometric equation is Cos(4x) = -1 + 2Cos^2(2x).