Question

Sin4x+cos4x+2sin2xcos2x=

Answer 1

Ответ:

2sin2xcos2x=sin4x

2sin4x+cos4x

Answer 2

To simplify the expression sin(4x) + cos(4x) + 2sin(2x)cos(2x), we can utilize the trigonometric identities to rewrite and simplify the terms.

Let's start by using the double-angle formulas: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x).

Substituting these identities into the expression, we get: sin(4x) + cos(4x) + 2sin(2x)cos(2x) = sin(4x) + cos(4x) + 2(2sin(x)cos(x))(cos²(x) - sin²(x)) = sin(4x) + cos(4x) + 4sin(x)cos(x)(cos²(x) - sin²(x)) = sin(4x) + cos(4x) + 4sin(x)cos(x)cos²(x) - 4sin(x)cos(x)sin²(x))

Now, we can apply the Pythagorean identity sin²(x) + cos²(x) = 1 to simplify the terms further: sin(4x) + cos(4x) + 4sin(x)cos(x)cos²(x) - 4sin(x)cos(x)sin²(x) = sin(4x) + cos(4x) + 4sin(x)cos(x)(1 - sin²(x)) - 4sin(x)cos(x)sin²(x) = sin(4x) + cos(4x) + 4sin(x)cos(x) - 4sin(x)cos(x)sin²(x) - 4sin(x)cos(x)sin²(x) = sin(4x) + cos(4x) + 4sin(x)cos(x) - 8sin(x)cos(x)sin²(x)

Finally, we can use the double-angle formula again to simplify the remaining terms: sin(4x) = 2sin(2x)cos(2x).

Replacing sin(4x) with 2sin(2x)cos(2x), we obtain: 2sin(2x)cos(2x) + cos(4x) + 4sin(x)cos(x) - 8sin(x)cos(x)sin²(x).

Therefore, the simplified expression is: 2sin(2x)cos(2x) + cos(4x) + 4sin(x)cos(x) - 8sin(x)cos(x)sin²(x).