Esercizio 17 
. Semplificare la seguente espressione:
![\[\dfrac{\cos\left(\alpha+\dfrac{\pi}{4}\right)\cdot\cos\left(\alpha-\dfrac{\pi}{4}\right)}{\cos(2\alpha)}.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-f89cfa848b1810456303d0425ee17504_l3.png "Rendered by QuickLaTeX.com")
Svolgimento. Per prima cosa applichiamo l’opportuna formula di Werner al numeratore:
![\[\begin{aligned} &\cos\left(\alpha+\dfrac{\pi}{4}\right)\cdot\cos\left(\alpha-\dfrac{\pi}{4}\right)=\\ &=\dfrac {1}{2} \left( \cos\left( \left(\alpha+\dfrac{\pi}{4}\right)+\left(\alpha-\dfrac{\pi}{4}\right)\right) + \cos\left(\left(\alpha+\dfrac{\pi}{4}\right)-\left(\alpha-\dfrac{\pi}{4}\right)\right)\right)=\\ &=\dfrac {1}{2} \left(\cos(2\alpha)+\cos\left(\dfrac{\pi}{2}\right)\right)= \dfrac {1}{2}\left(\cos(2\alpha)+0\right)=\dfrac{\cos(2\alpha)}{2}, \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-b15409fa819a0143941da873390b0f41_l3.png "Rendered by QuickLaTeX.com")
e sostituendo nell’espressione iniziale otteniamo
![\[\dfrac{\cos\left(\alpha+\dfrac{\pi}{4}\right)\cdot\cos\left(\alpha-\dfrac{\pi}{4}\right)}{\cos(2\alpha)}= \dfrac{\dfrac{\cos(2\alpha)}{2}}{\cos(2\alpha)}= \dfrac{\cos(2\alpha)}{2\cos(2\alpha)}=\dfrac{1}{2}.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-76bfc56623c996bb37872c2585b5c3a0_l3.png "Rendered by QuickLaTeX.com")
Fonte: Corso base blu di matematica di Massimo Bergami-Anna Trifone-Graziella Barozzi