Simplify 2cos2xsin2x ?? how to do

Question

i have to use trig identities to solve, confused.

Emery Adkins · Accepted Answer

You can simplify the expression to #sin(4x)#.

If we let #u=2x#, we can use the sine double angle formula backward: #color(white)=2sin(2x)# #cos(2x)# #=2sin(u)# #cos(u)# #=sin(2u)# #=sin(2*u)# #=sin(2*2x)# #=sin(4x)# That's as simplified as it gets. Hope this helped!

Adam Aguirre · Answer

undefined To simplify $2\cos(2x)\sin(2x)$, you can use the double angle identities for cosine and sine:

$$\cos(2x) = 2\cos^2(x) - 1$$
$$\sin(2x) = 2\sin(x)\cos(x)$$

Substitute these identities into the expression:

$$2(2\cos^2(x) - 1)(2\sin(x)\cos(x))$$

Expand and simplify:

$$4\cos^2(x) \cdot 2\sin(x)\cos(x) - 2(2\sin(x)\cos(x))$$

$$8\cos^2(x)\sin(x)\cos(x) - 4\sin(x)\cos(x)$$

$$4\sin(x)\cos(x)(2\cos^2(x) - 1)$$

Now, you can use the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to replace $\cos^2(x)$ with $1 - \sin^2(x)$:

$$4\sin(x)\cos(x)(2(1 - \sin^2(x)) - 1)$$

$$4\sin(x)\cos(x)(2 - 2\sin^2(x) - 1)$$

$$4\sin(x)\cos(x)(1 - 2\sin^2(x))$$

Finally, you can use the double angle identity for sine $\sin(2x) = 2\sin(x)\cos(x)$ to replace $2\sin(x)\cos(x)$ with $\sin(2x)$:

$$4\sin(2x)(1 - 2\sin^2(x))$$

So, $2\cos(2x)\sin(2x)$ simplifies to $4\sin(2x)(1 - 2\sin^2(x))$.