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(A) $\sin(-x)=-\sin x$
(B) $\cos(-x)=\cos x$
(C) $\cos(x+y)=\cos x\cos y-\sin x\sin y$
(D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$
Use these equalities to derive the following important trigonometric functions:
(b) $\sin 2x=2\sin(x)\cos(x)$

I know I've asked this same question about a month ago, but I just got back to this problem in my packet and I have yet to come up with anything. For (a), I may have gotten it. So please check here. I moved on to the second one and I have no idea how to start this one off. It's just a confusing question I think. Can someone at least tell me which equation(s) to use and why? Thanks a lot.

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    I believe from the wording that you are not expected to *prove* these facts. In particular, (C) and (D) are harder than other things you have been asking about. I think you are just supposed to use facts chosen from (A) to (D) to prove (b). This is easy.2012-08-10
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    Right, so for the next one, (e) $\cos2x=1-2\sin^2x$ (using d then a & $y=x$); $\sin(x+y)=\sin x\cos y+\sin y\cos x\Rightarrow \sin2x=-(-\sin x\cos x)+\cos x\sin x\Rightarrow \sin2x=\sin(-x)\cos x\sin x \cdots$. I believe I messed this one up horribly...2012-08-10
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    Whoa, what? I never found such problem in (e). Oh, wait. You want me to start off using (C) instead of (D)?2012-08-10
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    You **already** showed $\cos 2x=\cos^2 x-\sin^2 x$ using (C). Now it's just replacing $\cos^2 x$ by $1-\sin^2 x$.2012-08-10
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    I figured out (e). Now for the slightly difficult looking ones, (f) and (g). I'll start with (f); $\left|\cos\dfrac{x}{2}\right|=\sqrt{\dfrac{1+\cos x}{2}}$ Since this is a half-angle identity I replace $x$ with $\frac{\pi}{2}$. And I'll use (C). $\cos(\frac{\pi}{2}+\frac{\pi}{2})=\cos\frac{\pi}{2}\cos\frac{\pi}{2}-\sin\frac{\pi}{2}\sin\frac{\pi}{2}\Rightarrow \cos2\frac{\pi}{2}=\cos^2\frac{\pi}{2}-\sin^2\frac{\pi}{2}$ Using power reduction identity of: $\cos^2\theta=\dfrac{1+\cos2\theta}{2}$ yields $\cos2\frac{\pi}{2}=\dfrac{1+\cos2\frac{\pi}{2}}{2}$. I believe I messed this one up as well...2012-08-10
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    Ask it as a question and I or someone else will answer it. Awkward to do the typing in a comment.2012-08-10
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    You mean post a new question with this and (g)?2012-08-10
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    Yes, anything that takes space in typing is too hard to do in comment. One doesn't get feedback on the LaTeX, a little LaTeX error can make an awful mess.2012-08-10

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