Докажете тъждеството

[Гост]

Докажете тъждеството:

[Hephaestus]

Нека [tex]A = \left( cos^{2 } \frac{ \alpha }{2} + 2 sin \frac{ \alpha }{2} cos \frac{ \alpha }{2} - sin^{2} \frac{ \alpha }{2} \right)^{2 } - 1 \hspace{0.1cm}[/tex] и [tex]\hspace{0.1cm} B = sin 2 \alpha[/tex]

Ще преобразуваме [tex]A[/tex]:

[tex]A = \left( cos^{2 } \frac{ \alpha }{2} + 2 sin \frac{ \alpha }{2} cos \frac{ \alpha }{2} - sin^{2} \frac{ \alpha }{2} \right)^{2 } - 1 =[/tex]

[tex]= \left( cos^{2 } \frac{ \alpha }{2} - sin^{2} \frac{ \alpha }{2} + 2 sin \frac{ \alpha }{2} cos \frac{ \alpha }{2} \right)^{2 } - 1 =[/tex]

[tex]= (cos \alpha + sin \alpha)^{2} - 1 =[/tex]

[tex]= cos^{2} \alpha + 2sin\alpha cos\alpha + sin^{2}\alpha - 1 =[/tex]

[tex]=sin^{2}\alpha + cos^{2} \alpha - 1 + sin 2\alpha =[/tex]

[tex]=1 - 1 + sin 2\alpha =[/tex]

[tex]=sin 2\alpha \hspace{0.2cm} \Rightarrow \hspace{0.2cm} A = B \hspace{0.2cm} \Rightarrow \hspace{0.2cm}[/tex] тъждеството е доказано.