Se cunoaste fromula ctgu=[tex] \frac{cosu}{sinu} [/tex]
Deci ctg([tex] \frac{ \pi }{4} + \frac{x}{2} [/tex])=[tex] \frac{cos( \frac{ \pi }{4} + \frac{x}{2} )}{sin( \frac{ \pi }{4}+ \frac{x}{2} )} [/tex]
Se aplica formulele pt [tex]cos( \alpha + \beta ) si sin( \alpha + \beta )[/tex]
Vei avea:
[tex] \frac{( cos\frac{ \pi }{4} cos \frac{x}{2} )- ( sin\frac{ \pi }{4} sin \frac{x}{2} ) }{( sin\frac{ \pi }{4} cos \frac{x}{2} )- ( cos\frac{ \pi }{4} sin\frac{x}{2} )} [/tex]
Inlocuind valorile pt cos si sin pt[tex] \frac{ \pi }{4} [/tex]
in final iti va ramane:
[tex] \frac{cos \frac{x}{2}-sin \frac{x}{2} }{cos \frac{x}{2}+sin \frac{x}{2} } [/tex]
Vei amplifica cu numaratorul si iti va da:
[tex] \frac{ ( cos \frac{x}{2}- sin\frac{x}{2})^{2} }{ (cos \frac{x}{2})^{2}- (sin\frac{x}{2})^2}= \frac{1-sinx}{cosx} [/tex]
Deoarece s-au aplicat formulele pt cos2x si sin2x si formula [tex] cos^{2}x + sin^{2}x=1 [/tex]
