Bonsoir
1) [tex]cos^2x+sin^2x=1\Longrightarrow cos^2x=1-sin^2x[/tex]
et
[tex]cos^2x+sin^2x=1\Longrightarrow sin^2x=1-cos^2x[/tex]
Donc :
[tex]cos^2x-sin^2x=(1-sin^2x)-sin^2x=1-2sin^2x[/tex]
et
[tex]cos^2x-sin^2x=cos^2x-(1-cos^2x)=cos^2x-1+cos^2x=2cos^2x-1[/tex]
2) [tex]tan\ x=\dfrac{sinx}{cosx}\Longrightarrow tan^2\ x=\dfrac{sin^2x}{cos^2x}[/tex]
Donc
[tex]1+tan^2x=1+\dfrac{sin^2x}{cos^2x}\\\\1+tan^2x=\dfrac{cos^2x}{cos^2x}+\dfrac{sin^2x}{cos^2x}\\\\1+tan^2x=\dfrac{cos^2x+sin^2x}{cos^2x}\\\\1+tan^2x=\dfrac{1}{cos^2x}[/tex]