Odvoď vzorec pre tangens
Odvoď vzorec pre \( \large \tg \Large \frac{x}{2}\large \).
\( \large\frac{1\ -\ \textrm{\large cos}\ x}{\textrm{\large sin}\ x}=\frac{1\ -\ \textrm{\large cos}\ x}{\textrm{\large sin}\ x}\cdot\frac{1\ +\ \textrm{\large cos}\ x}{1\ +\ \textrm{\large cos}\ x}= \)
\( \large=\frac{1\ –\ \textrm{\large cos}^2\ x}{\textrm{\large sin}\ x\cdot(1+\textrm{\large cos}\ x)}=\frac{\textrm{\large sin}^2}{\textrm{\large sin}\ x\cdot(1+\textrm{\large cos}\ x)}= \)
\( \large =\Large \frac{\sin x}{1+\cos x}\large \)
\( \large\frac{\textrm{\large sin}\ x}{1\ -\ \textrm{\large cos}\ x}=\frac{\textrm{\large sin}\ x}{1\ -\ \textrm{\large cos}\ x}\cdot\frac{1\ +\ \textrm{\large cos}\ x}{1\ +\ \textrm{\large cos}\ x}= \)
\( \large=\frac{\textrm{\large sin}^2\ x}{1\ -\ \textrm{\large cos}^2\ x}=\frac{\textrm{\large sin}^2}{1\ -\ \textrm{\large cos}^2\ x}= \)
\( \large =\Large \frac{\sin x}{1-\cos x}\large \)
\( \large\frac{1\ -\ \textrm{\large cos}\ x}{\textrm{\large sin}\ x}=\frac{1\ -\ \textrm{\large cos}\ x}{\textrm{\large sin}\ x}\cdot\frac{1\ -\ \textrm{\large cos}\ x}{1\ -\ \textrm{\large cos}\ x}= \)
\( \large=\frac{1\ –\ \textrm{\large cos}^2\ x}{\textrm{\large sin}\ x\cdot(1-\textrm{\large cos}\ x)}=\frac{\textrm{\large sin}^2}{\textrm{\large sin}\ x\cdot(1-\textrm{\large cos}\ x)}= \)
\( \large =\Large \frac{\sin x}{1-\cos x}\large \)
\( \large\frac{1\ +\ \textrm{\large cos}\ x}{\textrm{\large sin}\ x}=\frac{1\ +\ \textrm{\large cos}\ x}{\textrm{\large sin}\ x}\cdot\frac{1\ -\ \textrm{\large cos}\ x}{1\ -\ \textrm{\large cos}\ x}= \)
\( \large=\frac{1\ –\ \textrm{\large cos}^2\ x}{\textrm{\large sin}\ x\cdot(1-\textrm{\large cos}\ x)}=\frac{\textrm{\large sin}^2}{\textrm{\large sin}\ x\cdot(1-\textrm{\large cos}\ x)}= \)
\( \large =\Large \frac{\sin x}{1-\cos x}\large \)