Question

Demonstrati:

1-sinx/cosx=ctg(\pi/4+x/2)

Answer 1

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