Can I prove sin(3t)+sin(t)=2sin(2t)cos(t) with these tools? (t=theta)

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Four Muffins

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Hello. I'm trying to prove trig identities, and I've gotten stuck on question 57 because I don't know how, or if I can, deal with a triple angle with what I have available. I realise I could probably solve it by Googling the triple angle formula, but I think that is not the intent of the exercise. I don't yet know how to derive the addition formula to get at the double or triple angle formula, and the exercise for that is 30 questions ahead of me.

I suspect I'm supposed to be able to solve this with what I've done so far, but I'm unsure. Either way, I've tried a bunch of times and always get stuck at either turning the right side into a triple angle, or turning the triple angle into something else.

I included the textbook material and previous questions for context.

These identities come with explanations, I chopped them out for space. The rest of the trig section is the functions and their graphs. It's only eight pages all up.
identities.png

The question I'm stuck on is 57. I haven't tried 58 yet, but I will have the same problem I bet.
questions.png

My latest two attempts. They all end in the same spot, with me trying a different way to manipulate [imath]cos^2(theta)[/imath]. I don't see another point at which to attack the problem after rewriting [imath]sin(2theta)[/imath]
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You know how to deal with trig functions where the angle is a sum, like sin( 135 + 315). Just think of 3theta as the sum of 2 other angles. See if this hint get you anywhere. If not, then come back and I'll tell you how to break up 3theta.
 
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Try splitting the \(\displaystyle 3\theta\) into \(\displaystyle 2\theta +\theta\).

Edit: Sorry Steven G, I gave your game away!
 
There's that thing about math always being so obvious right after it needs to be. :) With your hints I solved them both without trouble, thank you.
 
Four Muffins said:
There's that thing about math always being so obvious right after it needs to be. :) With your hints I solved them both without trouble, thank you.
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Always try to use what you already know in solving a problem that looks different to you. Also use definitions!

You can use imaginary number to compute sin(3theta) but maybe you haven't learned that yet.
 
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