Question

$\sin A+2\sin 2A+\sin 3A$ is equal to which of the following?
1. $4\sin 2A{\cos }^2\left(\frac{A}{2}\right)$
2. $2\sin 2A{(\sin \frac{A}{2}+\cos \frac{A}{2})}^2$
3. $8\sin A\cos A{\cos }^2\left(\frac{A}{2}\right)$
Select the correct answer using the code given below

• A. 1 and 2 only
• B. 2 and 3 only
• C. 1 and 3 only
• D. 1, 2 and 3

Answer 1

Answer: (C)

1 and 3 only

$\sin A+2\sin 2A+\sin 3A$

$=(\sin A+\sin 3A)+2\sin 2A$

$=\left(2\sin \right(2A\left)\cos \right(-A\left)\right)+2\sin 2A$

$=2\sin \left(2A\right)(\cos A+1)$ ...(1)

$=2\sin \left(2A\right)2{\cos }^2\left(\frac{A}{2}\right)$

$=4\sin 2A{\cos }^2\left(\frac{A}{2}\right)$ ...(2)

$=4\left(2\sin A\cos A\right){\cos }^2\left(\frac{A}{2}\right)$

$=8\sin A\cos A{\cos }^2\left(\frac{A}{2}\right)$ ...(3)

Expression II is $2\sin 2A{(\sin \frac{A}{2}+\cos \frac{A}{2})}^2$

$=2\sin 2A({\sin }^2\frac{A}{2}+{\cos }^2\frac{A}{2}+2\sin \frac{A}{2}\cos \frac{A}{2})$

$=2\sin 2A(1+sinA)$ ...(4)

We can see that (1)$\ne$(4), so expression II is wrong

From (2) and (3), expression I and III are true

Answer is option (C)