Which expression is equivalent to sin(3x) + sin x?
A) 2cos(2x)sin x
B) 2sin(2x)sin x
C) -2sin(2x)cos x
D) 2sin(2x)cos x
E) -2cos(2x)sin x
well, you have the formula
sin(u)+sin(v) = 2 sin (u+v)/2 cos (u-v)/2
So plug in u=3x and v=x
To figure out which expression is equivalent to sin(3x) + sin(x), we can use the trigonometric identity called the sum-to-product identity.
The sum-to-product identity states that sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2).
Now let's match this identity with the given expression:
sin(3x) + sin(x)
We can think of sin(3x) as sin(A) and sin(x) as sin(B). By comparing with the sum-to-product identity, we have:
A = 3x and B = x.
Now we can substitute these values into the sum-to-product identity:
sin(3x) + sin(x) = 2sin((3x + x)/2)cos((3x - x)/2)
Simplifying the equation:
= 2sin(4x/2)cos(2x/2)
= 2sin(2x)cos(x)
So the expression that is equivalent to sin(3x) + sin(x) is 2sin(2x)cos(x).
Therefore, the correct answer is option D) 2sin(2x)cos(x).