Question

Aratati ca (1-sinα)x²-2xcosα+1+sinα≥0 , ∀ x,d ∈ R

Answer 1

Avem $\sin \alpha=2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}$
și $\cos\alpha=\cos^2\frac{\alpha}{2}-\sin^2\frac{\alpha}{2}$
și $\sin^2\frac{\alpha}{2}+\cos^2\frac{\alpha}{2}=1$
Atunci expresia se scrie
$\left(\sin\frac{\alpha}{2}-\cos\frac{\alpha}{2}\right)^2x^2-2\left(\cos^\frac{\alpha}{2}-\sin^\frac{\alpha}{2}\right)+\left(\sin\frac{\alpha}{2}+\cos\frac{\alpha}{2}\right)^2$
care se restrânge la
$\left(\left(\sin\frac{\alpha}{2}-\cos\frac{\alpha}{2}\right)x+\left(\sin\frac{\alpha}{2}+\cos\frac{\alpha}{2}\right)\right)^2\geq 0$