00:01
So for this question, we're asked to prove that sine x plus sine 3x is equal to 4 sine x cosine squared x.
00:19
So on the left side, we can combine this sum and make it into product, using the sum to product formula, which states that if we have a sign, maybe alpha plus a sign of beta.
00:36
Then we have two sign alpha plus beta over two.
00:43
So alpha plus beta is x plus 3x, 4x over 2 is 2x, times cosine of alpha minus beta over 2.
00:57
So x minus 3x over 2.
01:05
Negative x and because cosine is an even function so that means cosine negative x is equal to cosine of x and you can prove this by you know looking at the unit circle and you'll notice that if you were to sort of travel along the unit circle in a positive direction so going from like zero to 30 degrees to 45 degrees those numbers exactly correspond to if you traveled in the negative directions.
01:38
If you went negative 30 degrees, negative 45 degrees, and so on.
01:43
So we know this to be true.
01:45
So we can go ahead and just remove, oh, that's not what i'm going to do, remove the negative sign...