Answer:
B. 2cos(3π/2)sin(π/2)
Explanation:
You want an equivalent expression for sin(2x) -sin(x) in terms of 3π/2x and π/2x.
Identities
The identities for sum and difference can be used:
sin(a+b) = sin(a)cos(b) +sin(b)cos(a)
sin(a-b) = sin(a)cos(b) -sin(b)cos(a)
Application
sin(2x) = sin(3πx/2 +πx/2) = sin(3πx/2)cos(πx/2) +sin(πx/2)cos(3πx/2)
sin(x) = sin(3πx/2 -πx/2) = sin(3πx/2)cos(πx/2) -sin(πx/2)cos(3πx/2)
The difference of these is ...
sin(2x) -sin(x) = sin(3πx/2)cos(πx/2) +sin(πx/2)cos(3πx/2) -(sin(3πx/2)cos(πx/2) -sin(πx/2)cos(3πx/2))
The first terms cancel, so the result is ...
sin(2x) -sin(x) = 2cos(3πx/2)sin(πx/2) . . . . . . . matches choice B
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Additional comment
You will notice that choices A and C are identical, so both can be eliminated from consideration.
We find a graphing calculator a reasonably quick way to tell if two functions are the same. The attachment shows the choices (solid lines) relative to the given expression (dots). This confirms that B (green) is a match.
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