[tex]sin^21^o + sin^22^o+...+sin^290^o= \\ \\ =sin^2 1^o+sin^22^o+...+sin^2 44^o+sin^245^o+ sin^2 46^o+...+sin^2 90^o = \\ \\ =sin^2 1^o+sin^2 2^o+...+sin^2 44^o+sin^2 45^o+(1-cos^{}^2 46^o)+...+sin^290^o= \\ \\ =sin^2 1^o+sin^2 2^o+...+sin^2 45^o+(1-cos^2(90^o-44^o))+...+sin^290^o = \\ \\ =sin^21^o+sin^22^o+...+sin^245^o+(1-sin^244^o)+(1-sin^243^o)+...+ \\ \\ +(1-sin^21^o)+sin^290^o = [/tex]
[tex]=sin^21^o+sin^22^o+...+sin^245^o-sin^244^o-sin^243^o-sin^242^o-...-sin^21^o+ \\ \\ +\underset{de \quad 44\quad ori}{1+1+...+1}+sin^290^o = sin^245^o+44+sin^290^o = \Big( \dfrac{ \sqrt{2} }{2} \Big)^2+44+1^2 = \\ \\ = \dfrac{2}{4} +45 = \dfrac{1}{2}+45 = \dfrac{1+90}{2} = \dfrac{91}{2} \\ \\ \\ $M-am folosit de proprietaile:: $
\left| \begin{array}{c} sin^2x = 1-cos^2x\\ cos(90^o-x) = sin x \end{array} \right[/tex]
