How to prepare myself for an advanced trignonometry exam

I'm gonna have a trigonometry/general algebra exam soon. My teacher has told us about some trignometric proofs, and we defined the $\sin$ and $\cos$ int he right way, doing all formal proofs for the formulas $\sin(a\pm b)$ and $\cos(a\pm b)$ for all $a,b\in\mathbb R$.

We'll have to prove some trig identities. That's where I think she'll make us prove some hard exercises. Like, identities that i've never seen.

I've had a sneak peek into a later exam she gave the class, and there was a identity like

$$\sin 2A + \sin 2B + \sin 2C = 4\sin(A)\sin(B)\sin(C)$$

When $A,B,C$ are angles of a triangle.

This is not an identity easy to prove at the middle of an exam, if you never seen it. I don't have enough time to think about it, there are many substitutions I need to make, in order to find the exact one that'll work.

So, how to protect myself against surprises in this test? Does somebody has an idea? Is there a book with these secret identities, that I can try to prove by myself at home?

Can you think of a good exercise or identity that you think will help me in the test?

• I think the book you're looking for is Gelfand & Saul, Trigonometry. – Marius Kempe Nov 20 '14 at 18:10

Besides identities, trigonometry covers some other topics such as (linear) equations (in the sine and cosine), derivatives of direct and inverse trigonometric functions that, depending on the exam, you should know.

To prove identities sometimes it is useful to manipulate the entire identity than transform the left hand side into the right hand one: see an example here.

In addition to the triangle sin and cos laws, it may well be that you need also to know the law of tangents.

The Wikibook's Trigonometry/For Enthusiasts/Less-Used Trig Identities page lists 8 difficult triangle identities, e. g. the 5th and 6th

$$\cot A \cot B + \cot B \cot C + \cot C \cot A = 1.$$

$$\cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2} = \cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2}.$$

The identity you state in the question has already been asked and answered in Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle.

Another set of triangle identities that are worth knowing: the following formula and similar for $B$ and $C$ due to H. Briggs, where $p=(a+b+c)/2$:

$$\tan\frac{A}{2}=\sqrt{\frac{(p-b)(p-c)}{p(p-a)}}.$$

If you have a text book, study it and try to solve the exercises and problems.