Sin2x+2sin2x+3cos2x=0

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вопрос

To simplify the given expression, we can start by applying the double angle formula for sine and cosine:

sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x)

Now we can substitute these expressions into the given equation:

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

Rearranging terms, we get:

2sin(x)cos(x) + 4sin(x)cos(x) + 3 - 6sin^2(x) = 0

Combining like terms:

6sin(x)cos(x) - 6sin^2(x) + 3 = 0

Dividing the entire equation by 3:

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

Using the double angle formula for sine once again:

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

sin^2(x) + cos^2(x) = 1

Therefore, the simplified expression is:

1 = 1

So, the given equation simplifies to 1 = 1, which is always true.