Esercizio 8 
. Semplificare la seguente espressione:
![\[\sin\left(\dfrac{\pi}{3}-\alpha\right)-\cos\left(\frac{\pi}{6}-\alpha\right).\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-6df20fd1fd49b5a7214c3b1c9bc2b8a7_l3.png "Rendered by QuickLaTeX.com")
Svolgimento. Possiamo sviluppare entrambi i termini dell’espressione utilizzando le formule di addizione e sottrazione, in particolare si ha:
![\[\sin\left(\dfrac{\pi}{3}-\alpha\right)=\sin\left(\dfrac{\pi}{3}\right)\cos(\alpha)-\cos\left(\dfrac{\pi}{3}\right)\sin(\alpha)= \dfrac{\sqrt{3}}{2}\cos(\alpha)-\dfrac{1}{2}\sin(\alpha)\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-c7b2f6abcf61212b2a42a2ee95899f81_l3.png "Rendered by QuickLaTeX.com")
e
![\[\cos\left(\dfrac{\pi}{6}-\alpha\right)=\cos\left(\dfrac{\pi}{6}\right)\cos(\alpha)+\sin\left(\dfrac{\pi}{6}\right)\sin(\alpha)= \dfrac{\sqrt{3}}{2}\cos(\alpha)+\dfrac{1}{2}\sin(\alpha),\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-fb5dd0a14890612a8b607dfd829d2959_l3.png "Rendered by QuickLaTeX.com")
da cui
![\[\begin{aligned} \sin\left(\dfrac{\pi}{3}-\alpha\right)-\cos\left(\frac{\pi}{6}+\alpha\right)&= \dfrac{\sqrt{3}}{2}\cos(\alpha)-\dfrac{1}{2}\sin(\alpha)- \left(\dfrac{\sqrt{3}}{2}\cos(\alpha)+\dfrac{1}{2}\sin(\alpha)\right)=\\ &= \dfrac{\sqrt{3}}{2}\cos(\alpha)-\dfrac{1}{2}\sin(\alpha)- \dfrac{\sqrt{3}}{2}\cos(\alpha)-\dfrac{1}{2}\sin(\alpha)=\\ &=-\sin\left(\alpha\right), \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-12721e537604d53acdb003dedd273624_l3.png "Rendered by QuickLaTeX.com")
ottenendo così la semplificazione richiesta.
Fonte: Corso base blu di matematica di Massimo Bergami-Anna Trifone-Graziella Barozzi