Sin2x+√2sin(4x+п/4)=cos4x

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To solve the equation sin(2x) + √2sin(4x + π/4) = cos(4x), we can use trigonometric identities to simplify the equation.

First, let's simplify the terms using the trigonometric identity sin(2A) = 2sin(A)cos(A) and cos(2A) = 1 - 2sin^2(A):

sin(2x) + √2sin(4x + π/4) = cos(4x)
2sin(x)cos(x) + √2sin(4x)cos(π/4) = 1 - 2sin^2(2x)

Now simplify further:

2sin(x)cos(x) + √2sin(4x)cos(π/4) = 1 - 2(2sin(x)cos(x))^2
2sin(x)cos(x) + √2sin(4x)cos(π/4) = 1 - 8sin^2(x)cos^2(x)

Since sin(π/4) = cos(π/4) = 1/√2, we can simplify further:

2sin(x)cos(x) + √2sin(4x)(1/√2) = 1 - 8sin^2(x)cos^2(x)
2sin(x)cos(x) + sin(4x) = 1 - 8sin^2(x)cos^2(x)

Now we need to expand the terms and simplify:

2sin(x)cos(x) + 2sin(2x)cos(2x) = 1 - 8(sin^2(x))(1 - sin^2(x))
2sin(x)cos(x) + 2sin(2x)cos(2x) = 1 - 8(sin^2(x) - sin^4(x))
2sin(x)cos(x) + 2sin(2x)cos(2x) = 1 - 8sin^2(x) + 8sin^4(x)

Now we can rewrite sin(2x) = 2sin(x)cos(x) and sin(4x) = 2sin(2x)cos(2x):

sin(2x) + sin(4x) = 1 - 8sin^2(x) + 8sin^4(x)

This equation can be further simplified if needed for a specific solution.