Question

VA ROOOOOOG SA MA AJUTATI OFER COROANA!

Answer 1

   
[tex]\displaystyle \\ \texttt{Folosim formula:} \\ \\ tg \frac{x}{2} =\pm \sqrt{ \frac{1-\cos x}{1+\cos x} } ~~~\Longrightarrow ~~ tg^2 \frac{x}{2} = \frac{1-\cos x}{1+\cos x} \\ \\ \\ \texttt{Rezolvare: } \\ \\ \\ tg^2 \frac{x_1}{2} + tg^2 \frac{x_2}{2}+ \cdots + \frac{x_n}{2} = \\ \\ = \frac{1-\cos x_1}{1+\cos x_1} + \frac{1-\cos x_2}{1+\cos x_2} +\cdots + \frac{1-\cos x_1}{1+\cos x_1} = \\ \\ ~~~~~~\Big(Dar~ \cos x_k = \frac{a_k}{S-a_k} \Big) \\ \\ [/tex]


[tex]\displaystyle \\ = \frac{1-\frac{a_1}{S-a_1}}{1+\frac{a_1}{S-a_1}} + \frac{1-\frac{a_2}{S-a_2}}{1+\frac{a_2}{S-a_2}} +\cdots + \frac{1-\frac{a_n}{S-a_n}}{1+\frac{a_n}{S-a_n}} = \\ \\ \\ = \frac{\frac{S-a_1-a_1}{S-a_1}}{\frac{S-a_1+a_1}{S-a_1}} + \frac{\frac{S-a_2-a_2}{S-a_2}}{\frac{S-a_2+a_2}{S-a_2}} +\cdots + \frac{\frac{S-a_n-a_n}{S-a_n}}{\frac{S-a_n+a_n}{S-a_n}} = [/tex]


[tex]\displaystyle \\ = \frac{\frac{S-2a_1}{S-a_1}}{\frac{S}{S-a_1}} + \frac{\frac{S-2a_2}{S-a_2}}{\frac{S}{S-a_2}} +\cdots + \frac{\frac{S-2a_n}{S-a_n}}{\frac{S}{S-a_n}} = \\ \\ \\ = \frac{S-2a_1}{S} +\frac{S-2a_2}{S} +\cdots +\frac{S-2a_n}{S} = \\ \\ = \frac{S-2a_1 +S-2a_2 +\cdots + S-2a_n }{S} = \\ \\ = \frac{n\times S-2(a_1 + a_2 +\cdots + a_n)}{S} = \\ \\ = \frac{n \times S-2 \times S}{S} = \frac{S (n-2)}{S} = \boxed{n-2} \\ \\ cctd [/tex]