Question 614139
These problems are meant to help you learn that the Trig properties/formulas/identities you've learned are patterns. The x's, A's and B's in the various formulas should be looked upon as placeholders. They can be replaced by <i>any</i> number, variable or expression and the equation will still be true!<br>
So when you look at
sin(2x) = 2sin(x)cos(x)
think "the sin of 2 times anything is equal to 2 times the sin of the anything times the cos of the anything (and vice versa)" Once you get comfortable with this you will start seeing that your problem #1:
2sin(4x)cos(4x)
exactly matches the pattern of the right side of sin(2x) with the "anything" being 4x. So according to the pattern this must be equal to the sin of 2 times the anything:
sin(2*(4x)) or sin(8x)<br>
#2. {{{2cos^2(3x) - 1}}}
This fits the pattern of the right side of:
{{{cos(2x) = 2cos^2(x) - 1}}}
with the "anything" being 3x. So according to the pattern the right side must be equal to the left side: the cos of 2 times the "anything":
cos(2(3x)) = cos(6x)<br>
#3 {{{1-2sin^2(4x)}}}
This fits the pattern of the right side of:
{{{cos(2x) = 1 - 2sin^2(x)}}}
In hopes that you've started understanding this, I'll leave it up to you to finish.