Which expression is equivalent to cos(4x) + cos(2x)?
cos(4x) + cos(2x)
=cos(3x+x)+cos(3x-x)
=2cos(3x)cos(x)
To determine an expression equivalent to cos(4x) + cos(2x), we can use the trigonometric identity:
cos(A) + cos(B) = 2 * cos((A + B) / 2) * cos((A - B) / 2)
In this case, let A = 4x and B = 2x. We can substitute these values into the identity:
cos(4x) + cos(2x) = 2 * cos((4x + 2x) / 2) * cos((4x - 2x) / 2)
Simplifying further:
cos(4x) + cos(2x) = 2 * cos(3x) * cos(x)
Therefore, the equivalent expression is 2 * cos(3x) * cos(x).
To find an expression equivalent to cos(4x) + cos(2x), we can use the trigonometric identity known as the angle sum formula for cosine. The formula states that:
cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
By applying this formula, we can rewrite cos(4x) + cos(2x) as follows:
cos(4x) + cos(2x) = cos(2x + 2x) + cos(2x)
Next, we can use the angle sum formula to break down cos(2x + 2x) into a combination of cosine and sine:
cos(2x + 2x) = cos(2x) * cos(2x) - sin(2x) * sin(2x)
Now, let's simplify:
cos(4x) + cos(2x) = (cos(2x) * cos(2x) - sin(2x) * sin(2x)) + cos(2x)
Since cos²(2x) - sin²(2x) is a well-known trigonometric identity, we can simplify further:
cos(4x) + cos(2x) = cos²(2x) - sin²(2x) + cos(2x)
Finally, combining the terms:
cos(4x) + cos(2x) = cos(2x) * (cos(2x) + 1) - sin²(2x)
Therefore, an equivalent expression to cos(4x) + cos(2x) is cos(2x) * (cos(2x) + 1) - sin²(2x).