[tex]Avem,sin \alpha +sin \beta =2sin \frac{ \alpha + \beta }{2}cos \frac{ \alpha - \beta }{2} [/tex], deci: sinx+cosx=sinx+sin(π/2-x)=
[tex]2[/tex][tex]sin \frac{x + \frac{ \pi }{2} -x}{2}*cos \frac{x- \frac{ \pi }{2}+x }{2} [/tex]=
2[tex]sin \frac{ \pi }{4}*cos(x- \frac{ \pi }{4})=2* \frac{ \sqrt{2} }{2}*cos(x- \frac{ \pi }{4})= \sqrt{2}cos(x- \frac{ \pi }{4}) [/tex]. Metoda a doua: dam factor fortat pe √2, si avem:
[tex]sinx+cosx=[/tex][tex] \sqrt{2}( [/tex][tex] \frac{ \sqrt{2} }{2}cosx+ \frac{ \sqrt{2} }{2}sinx)= \sqrt{2}(cosxcos \frac{ \pi }{4} +sinxsin \frac{ \pi }{4})= \sqrt{2}cos(x- \frac{ \pi }{4}) [/tex]
