Bases mathématiques

1. $\begin{align*} \cos^2 x+\sec^2 x \sin^2 x \cos^2 x &=1\\ \cos^2 x+\frac{1}{\cos^2 x}\sin^2 x \cos^2&=\\ \cos^2 x+\sin^2 x&=\\ 1&=\\ &\blacksquare \end{align*}$
2. $\begin{align*} \frac{\sin^2 x-\sin^4 x}{\cos^4 x-\cos^2 x}&=-1 \\ \frac{\sin^2 x\left(1-\sin^2 x\right)}{-\cos^2 x\left(-\cos^2 x+1\right)}&= \\ \frac{\sin^2 x \cos^2 x}{-\cos^2 x \sin^2 x}&= \\ -1&=\\ \blacksquare \end{align*}$
1. $\begin{align*} S=\{0\} \cup \left\{x\in \mathbb{R}|x=\frac{4Pi}{3}+n\pi, n\in\mathbb{N} \right\} \cup \left\{x\in \mathbb{R}|x=\frac{5Pi}{3}+n\pi, n\in\mathbb{N} \right\} \end{align*}$
$\begin{align*} \cos x=0 &\text{ ou }\tan x=0 \\ S=\left\{x\in \mathbb{R}|x=\frac{\pi}{2}+n\pi, n\in\mathbb{N} \right\} & \cup \left\{x\in \mathbb{R}|x=\frac{Pi}{4}+n\pi, n\in\mathbb{N} \right\} \end{align*}$

Exercices de vérification