(1)

２倍角の公式
\[ \cos2\frac{x}{2}=1-2\sin^{2}\frac{x}{2} \]
より、
\[ \sin^{2}\frac{x}{2}=\frac{1-\cos x}{2} \]

(2)

２倍角の公式
\[ \cos2\frac{x}{2}=2\cos^{2}\frac{x}{2}-1 \]
より、
\[ \cos^{2}\frac{x}{2}=\frac{1+\cos x}{2} \]

(3)

\begin{align*} \tan^{2}\frac{x}{2} & =\frac{\sin^{2}\frac{x}{2}}{\cos^{2}\frac{x}{2}}\\ & =\frac{1-\cos x}{1+\cos x} \end{align*}

(1)

\begin{align*} \sinh^{2}\frac{x}{2} & =-\sin^{2}\frac{ix}{2}\\ & =-\frac{1-\cos ix}{2}\\ & =\frac{\cosh x-1}{2} \end{align*}

(2)

\begin{align*} \cosh^{2}\frac{x}{2} & =\cos^{2}\frac{ix}{2}\\ & =\frac{1+\cos ix}{2}\\ & =\frac{\cosh x+1}{2} \end{align*}

(3)

\begin{align*} \tanh^{2}\frac{x}{2} & =\frac{\sinh^{2}\frac{x}{2}}{\cosh^{2}\frac{x}{2}}\\ & =\frac{\cosh x-1}{\cosh x+1} \end{align*}

• 三角関数の部分分数展開
  \[ \pi\tan\pi x =-\sum_{k=-\infty}^{\infty}\frac{1}{x+\frac{1}{2}+k} \]