Sinx +1 + sin5x = 2cos^2 x

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вопрос

To solve this equation, we will use trigonometric identities to simplify both sides of the equation.

We know that
sin^2 x + cos^2 x = 1

Starting with the left side of the equation
sinx + 1 + sin5
= sinx + sin5x +
= 2sin(3x)cos(2x) + 1

Now, we simplify the right side of the equation using the double angle identity for cosine
2cos^2
= 2(1 - sin^2 x
= 2 - 2sin^2 x

Substitute 2sin^2 x with 2 - 2cos^2 x in the equation
2sin(3x)cos(2x) + 1 = 2 - 2cos^2 x

Rearranging, we get
2sin(3x)cos(2x) = 1 - 2cos^2
sin(3x) = (1 - 2cos^2 x) / 2cos(2x)

This is as simplified as we can get without making approximations or giving a numerical solution.