[tex]\displaystyle\bf\\Cos\frac{7\pi }{5} +Cos\frac{9\pi }{5} =2Cos \ \frac{\dfrac{7\pi }{5} +\dfrac{9\pi }{5} }{2} \ Cos \ \frac{\dfrac{7\pi }{5} -\dfrac{9\pi }{5} }{2}=\\\\\\=2Cos\frac{16\pi }{10} Cos\frac{2\pi }{10} =2Cos\frac{8\pi }{5} Cos\frac{\pi }{5} =2Cos\Big(2\pi -\frac{2\pi }{5} \Big) Cos\frac{\pi }{5} =\\\\\\=2Cos\frac{2\pi }{5} Cos\frac{\pi }{5} =2\Big(2Cos^{2} \dfrac{\pi }{5} -1\Big)\cdot Cos\frac{\pi }{5} =4Cos^{3} \frac{\pi }{5} -2Cos\frac{\pi }{5} =[/tex]

[tex]\displaystyle\bf\\=4\Big(\frac{\sqrt{5} +1}{4} \Big)^{3} -2\cdot\frac{\sqrt{5} +1}{4} =\frac{1}{16} \Big(5\sqrt{5} +15+3\sqrt{5} +1\Big)-\frac{1}{2} \sqrt{5} -\frac{1}{2} =\\\\\\=\frac{1}{2} \sqrt{5} +1-\frac{1}{2} \sqrt{5} -\frac{1}{2} =\frac{1}{2} =0,5[/tex]

Дополнительное пояснение :

[tex]\displaystyle\bf\\Cos\frac{\pi }{5} =Cos36^\circ=\frac{\sqrt{5} +1}{4}[/tex]