Question

Which expression is equivalent to sin(2x) − sinx?

A. 2cos(x/2) sin(3x/2)
B. 2cos (3x/2) sin(x/2)
C. 2sin(3x/2) cos(x/2)
D. 2sin (3x/2) sin(x/2)

Answer 1

$\bf \textit{Sum to Product Identities} \\ \quad \\ sin({{ \alpha}})+sin({{ \beta}})=2sin\left(\cfrac{{{ \alpha}}+{{ \beta}}}{2}\right)cos\left(\cfrac{{{ \alpha}}-{{ \beta}}}{2}\right) \\ \quad \\\\ \boxed{sin({{ \alpha}})-sin({{ \beta}})=2cos\left(\cfrac{{{ \alpha}}+{{ \beta}}}{2}\right)sin\left(\cfrac{{{ \alpha}}-{{ \beta}}}{2}\right)} \\ \quad \\\\\ cos({{ \alpha}})+cos({{ \beta}})=2cos\left(\cfrac{{{ \alpha}}+{{ \beta}}}{2}\right)cos\left(\cfrac{{{ \alpha}}-{{ \beta}}}{2}\right) \\ \quad$

$\bf cos({{ \alpha}})-cos({{ \beta}})=-2sin\left(\cfrac{{{ \alpha}}+{{ \beta}}}{2}\right)sin\left(\cfrac{{{ \alpha}}-{{ \beta}}}{2}\right)\\\\ -------------------------------\\\\ sin(2x)-sin(x)\implies 2cos\left( \cfrac{2x+x}{2} \right)sin\left( \cfrac{2x-x}{2} \right) \\\\\\ 2cos\left( \cfrac{3x}{2} \right)sin\left( \cfrac{x}{2} \right)$

Answer 2

The expression sin(2x) − sinx is equivalent to 2sin (3x/2) sin(x/2) therefore option (D) is correct.

Recall the formula of sine

$\sin x-\sin y=2 \sin (\frac{x+y}{2}) \sin (\frac{x-y}{2})$

How to find the equivalent expression?

The given expression is sin (2x) − sin x.

Apply the formula

$\sin (2x) - \sin x =2 \sin (\frac{2x+x}{2}) \sin (\frac{2x-x}{2})$

                      $=2 \sin (\frac{3x}{2}) \sin (\frac{x}{2})$

Hence the expression is equivalent to 2sin (3x/2) sin(x/2)

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