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(Sin 11 pi/36)(cos pi/18) - (cos 11pi/36)(sin pi/18). I have to find the exact value

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From the trigonometric identity of the sum of angles


\sin (\alpha-\beta)=\sin \alpha\cos \beta-\cos \alpha\sin \beta

we ca note that


\begin{gathered} \alpha=(11\pi)/(36) \\ \beta=(\pi)/(18) \end{gathered}

Then, by substituting these values into the formula, we have


\sin ((11\pi)/(36)-(\pi)/(18))=\sin (11\pi)/(36)\cos (\pi)/(18)-\cos (11\pi)/(36)\sin (\pi)/(18)

which is the same given expression. Then, we have that


\sin ((11\pi)/(36)(\pi)/(18))=\sin ((11\pi)/(36)-(2\pi)/(36))=\sin ((11\pi)/(36)-(2\pi)/(36))=\sin (9\pi)/(36)=\sin (\pi)/(4)

Then, the answer is:


\sin (\pi)/(4)=\frac{\sqrt[]{2}}{2}

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