Verified answer

Первый способ :

[tex]\displaystyle\bf\\Cos\alpha +Sin\beta =Cos\alpha +Cos\Big(\frac{\pi }{2} -\beta \Big)=2Cos\frac{\alpha +\frac{\pi }{2} -\beta }{2} Cos\frac{a-\frac{\pi }{2} +\beta }{2}[/tex]

Второй способ :

[tex]\displaystyle\bf\\Cos\alpha +Sin\beta =Sin\Big(\frac{\pi }{2} -\alpha\Big )+Sin\beta =2Sin\frac{\frac{\pi }{2} -\alpha +\beta }{2} Cos\frac{\frac{\pi }{2}-\alpha -\beta }{2}[/tex]