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
Looks like D to me.
sin u + sin v = 2 sin(½(u+v)) cos(½(u−v))
To find which expression is equivalent to sin(3x) + sin x, we need to simplify it by applying trigonometric identities.
Starting with sin(3x) + sin x, we can use the trigonometric identity for the sum of two sines:
sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)
Using the identity, we can rewrite sin(3x) + sin x as:
2sin((3x + x)/2)cos((3x - x)/2)
Simplifying further, we have:
2sin(4x/2)cos(2x/2)
This can be simplified to:
2sin(2x)cos(x)
Therefore, the answer is D. 2sin(2x)cos(x).