Simplify the given expression........?

(2sin2x)(cos6x)

sin 2x and cos 6x can be expressed as a series of terms that involve sin x or cos x only, but the end result is not a simplification.
sin 2x = 2 sinx cosx
cos 6x = 32 cos^6 x -48 cos^4 x
+ 18 cos^2 x - 1

I assume you are not talking abous sin^2 x and cos^6 x

cos 6x could be written 2 cos^2 3x -1
but that doesn't help much, either

To simplify the expression (2sin2x)(cos6x), we can use the identity sin2x = 2sinxcosx.

We start by substituting sin2x with 2sinxcosx:
(2sinxcosx)(cos6x)

Now we can simplify further by applying the distributive property:
2sinxcosx * cos6x = (2cosx * cos6x) * sinx

Next, we can simplify the inside expression (2cosx * cos6x) by using the identity cos(a + b) = cosacosb - sinasinb. In this case, a = x and b = 6x:
2cosx * cos6x = cos(x + 6x) + cos(x - 6x)

Simplifying the sum inside the cosine function:
cos(x + 6x) = cos7x

And simplifying the difference inside the cosine function:
cos(x - 6x) = cos(-5x) = cos5x

Finally, we can rewrite our simplified expression as:
(2cos7x + 2cos5x) * sinx

So the simplified expression is (2cos7x + 2cos5x) * sinx.