Question

Verify the identity.

sin 4x = 2 sin 2x cos 2x

Answer 1

Start with sin(4x)=sin(2x+2x)

Answer 2

yeah.........whats next??

Answer 3

@campbell_st i don't understand..

Answer 4

Let \(y=2x\) \[\begin{split}
\sin (4x) &= \sin (2y) &\text{sub } y=2x\\
&= \sin (y+y) \\
&= \sin (y)\cos(y)+\sin(y) \cos (y)\\
&=2 \sin (y)\cos(y)&\text{sub }y=2x\\
&= 2 \sin (2x) \cos (2x)
\end{split}\]

Answer 5

ok... so

there is a trig identity called the sum of 2 angles

for sin its

\[\sin(a + b) = \sin(a)\cos(b) + \cos(a)(\sin(b)\]

you will need to use it.

so in your question split the 4x in 2 equal parts 2x and 2x

so

\[\sin(4x) = \sin(2x + 2x)\]

now using the expansion above you will get

\[\sin(2x + 2x) = \sin(2x)\times \cos(2x) + \cos(2x) \times \sin(2x)\]

it will simplify to

\[2\sin(2x)\cos(2x)\]

I hope this helps.

Answer 6

@campbell_st it helps so much thank you!!

Answer 7

glad it makes sense. good luck