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I have a question. How would I prove the following.

sin((2 arcsin(4/5)-arccos(12/13))=323/325

How would I solve this I have an idea I know

sin(a-b)=sin(a)cos(b)-cos(a)sin(b)

But I am not sure what to do.

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    Why not compute $\arcsin(4/5)$ and $\arccos(12/13)$, then substitute those values into the given expression?2012-12-14
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    that's awfully similar to your earlier post. Does anything you learned from that post help here? http://math.stackexchange.com/q/258289/90032012-12-14
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    There is no statement, and therefore nothing to "prove". Perhaps you mean that you want a way to rewrite it in a simpler form that no longer has trig and inverse trig functions?2012-12-14
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    I mean how would I make it equal 323/325 I do not know how to do it.2012-12-14
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    I know in the last problem someone offered me a formula to help solve the problem, but I am not sure what I would use here.2012-12-14

1 Answers 1

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Let $\arccos \frac{12}{13}=B$ and $\arcsin\frac45 =A$

$\displaystyle\implies 0\le B\le \pi$ and $\displaystyle-\frac\pi2\le A\le\frac\pi2$ (using the definition of principal value)

$\displaystyle\implies \sin B\ge0$ and $\displaystyle\cos A\ge0$

$\displaystyle\cos B=\frac{12}{13}\implies \sin B=+\sqrt{1-\left(\frac{12}{13}\right)^2}=+\frac5{13}$

and $\displaystyle\sin A=\frac45\implies \cos A=+\sqrt{1-\left(\frac45\right)^2}=+\frac35$

So, $$\sin(2A-B)=\sin 2A\cos B-\cos 2A\sin B$$ $$=2\sin A\cos A\cos B-(1-2\sin^2A)\sin B$$ $$=2\cdot\frac45\cdot \frac35\cdot\frac{12}{13}-\{1-2\left(\frac45 \right)^2\}\cdot \frac5{13} $$ $$=\frac{288-(-7)5}{13\cdot25}=\frac{323}{325}$$

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    thank you for your answer. But I have a quick question2012-12-14
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    How did you know sin2AcosB-cos2AsinB=2sinAcosAcosB-(1-2sin^2A)sinB.2012-12-14
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    @FernandoMartinez, are you aware of Multiple-Angle Formulas(http://mathworld.wolfram.com/Multiple-AngleFormulas.html)?2012-12-14