Тригонометријске функције двоструког угла 2

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Пр.5 $\frac{{t{g^2}\alpha  + ct{g^2}\alpha  - 6}}{{t{g^2}\alpha  + ct{g^2}\alpha  + 2}} = \cos 4\alpha $

Пр.6 $\frac{{2 - \sin 4\alpha tg2\alpha }}{{\sin 4\alpha }} = ctg2\alpha $

Пр.5

\[\begin{gathered}
  \frac{{t{g^2}\alpha  + ct{g^2}\alpha  - 6}}{{t{g^2}\alpha  + ct{g^2}\alpha  + 2}} = \cos 4\alpha  \hfill \\
  \frac{{\frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} + \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} - 6}}{{\frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} + \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2}} = \cos 4\alpha  \hfill \\
  \frac{{\frac{{{{\sin }^4}\alpha  + {{\cos }^4}\alpha  - 6{{\sin }^2}\alpha {{\cos }^2}\alpha }}{{{{\sin }^2}\alpha {{\cos }^2}\alpha }}}}{{\frac{{{{\sin }^4}\alpha  + {{\cos }^4}\alpha  + 2{{\sin }^2}\alpha {{\cos }^2}\alpha }}{{{{\sin }^2}\alpha {{\cos }^2}\alpha }}}} = \cos 4\alpha  \hfill \\
  \frac{{{{\sin }^4}\alpha  + {{\cos }^4}\alpha  - 6{{\sin }^2}\alpha {{\cos }^2}\alpha }}{{{{\left( {{{\sin }^2}\alpha  + {{\cos }^2}\alpha } \right)}^2}}} = \cos 4\alpha  \hfill \\
  {\sin ^4}\alpha  + {\cos ^4}\alpha  - 2{\sin ^2}\alpha {\cos ^2}\alpha  - 4{\sin ^2}\alpha {\cos ^2}\alpha  = \cos 4\alpha  \hfill \\
  {\left( {{{\cos }^2}\alpha  - {{\sin }^2}\alpha } \right)^2} - {\left( {2\sin \alpha \cos \alpha } \right)^2} = \cos 4\alpha  \hfill \\
  {\cos ^2}2\alpha  - {\sin ^2}2\alpha  = \cos 4\alpha  \hfill \\
  \cos 2 \cdot 2\alpha  = \cos 4\alpha  \hfill \\
  \cos 4\alpha  = \cos 4\alpha  \hfill \\
\end{gathered} \]

Пр.6   \[\begin{gathered}
  \frac{{2 - \sin 4\alpha tg2\alpha }}{{\sin 4\alpha }} = ctg2\alpha  \hfill \\
  \frac{{2 - \sin 2 \cdot 2\alpha  \cdot \frac{{\sin 2\alpha }}{{\cos 2\alpha }}}}{{\sin 2 \cdot 2\alpha }} = ctg2\alpha  \hfill \\
  \frac{{2 - 2\sin 2\alpha \cos 2\alpha  \cdot \frac{{\sin 2\alpha }}{{\cos 2\alpha }}}}{{2\sin 2\alpha \cos 2\alpha }} = ctg2\alpha  \hfill \\
  \frac{{2\left( {1 - {{\sin }^2}2\alpha } \right)}}{{2\sin 2\alpha \cos 2\alpha }} = ctg2\alpha  \hfill \\
  \frac{{{{\cos }^2}2\alpha }}{{\sin 2\alpha \cos 2\alpha }} = ctg2\alpha  \hfill \\
  \frac{{\cos 2\alpha }}{{\sin 2\alpha }} = ctg2\alpha  \hfill \\
  ctg2\alpha  = ctg2\alpha  \hfill \\
\end{gathered} \]