Formulaire : Trigonométrie

$$ \cos (-a) = \cos (a)$$ $$ \sin (-a)=-\sin (a)$$ $$ \tan (-a) =- \tan (a)$$

$$\cos (a+b) =\cos a \cos b-\sin a \sin b $$ $$\cos (a-b) =\cos a \cos b+\sin a \sin b $$ $$\sin (a+b) =\sin a \cos b+\cos a \sin b $$ $$\sin (a-b) =\sin a \cos b-\cos a \sin b $$ $$\tan (a+b) =\frac{\tan a+\tan b}{1-\tan a \tan b} $$ $$\tan (a-b) =\frac{\tan a-\tan b}{1+\tan a \tan b}$$ $$ \cos \alpha=\sin \left(\frac{\pi}{2}-\alpha\right) $$ $$ \sin \alpha=\cos \left(\frac{\pi}{2}-\alpha\right) $$

$$ \cos \alpha=\sin \left(\frac{\pi}{2}+\alpha\right) $$ $$ \sin \alpha=-\cos \left(\frac{\pi}{2}+\alpha\right) $$

$$\cos 2 a=\cos ^2 a-\sin ^2 a$$
$$\sin 2 a=2 \sin a \cos a$$
$$\tan 2 a=\dfrac{2 \tan a}{1-\tan ^2 a}$$

Comme $\cos ^2 a+\sin ^2 a=1$, alors $\cos ^2 a-\sin ^2 a=2 \cos ^2 a-1=1-2 \sin ^2 a$ et il vient :

$$
\begin{aligned}
\cos 2 a &=2 \cos ^2 a-1 \\
&=1-2 \sin ^2 a
\end{aligned}
$$