You can put this solution on YOUR website! One part of proving identities is to get the arguments to Trig functions on both sides to match. And changing arguments to Trig functions can only be done using a Trig identity.
To get good at using the various Trig identities you must learn that they are just patterns. For example:
sin(2x) = 2sin(x)cos(x)
What is critical here is that the argument on the left is twice the arguments on the right. (Or the arguments on the right are half of the argument on the left). So this identity can be used for any sin!!
sin(x) = 2sin(x/2)cos(x/2)
sin(1000q) = 2sin(500q)cos(500q)
etc.
So we can use sin(2x) on sin(4x):
We can use sin(2x), again, on sin(2x) and we can use cos(2x). (There are 3 variations of cos(2x). Any of them may be used. I will use (because I can see that it will make the rest of the problem a little easier):
All of this has been to get the argument(s) on the left, which started at 4x, to match the arguments on the right, x.
Now we simplify:
Distributing (sin(x)cos(x)):
And we're finished!
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