# 2 2arcsin x arcsin parenleftbig 2 x radicalbig 1 x 2

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2. 2Arcsin( x ) =Arcsin parenleftBig 2 x radicalbig 1 x 2 parenrightBig 3. Arccos( x )=Arcsin(2 x ) . 4. Arccos parenleftBig 1 x 1+ x parenrightBig +Arcsin parenleftBig 2 x 1+ x parenrightBig = π . Exercice 13 1. Montrer que pour tout x R , th( x )= 2 th(2 x ) 1 th( x ) . 2. En déduire la valeur de n summationdisplay k =0 2 k th(2 k x ) pour n N . Exercice 14 – Pour ( n, a, b ) N × R 2 , calculer n 1 summationdisplay k =0 ch( a + kb ) et n 1 summationdisplay k =0 sh( a + kb ) . Exercice 15 – Résoudre l’équation a ch( x )+ b sh( x )=0 ( ( a, b ) R 2 ), d’inconnue x dans R . Exercice 16 – Calculer la dérivée n -ième de f : x mapsto→ e x ch a ch( x sh( a )) , où a R est donné. 2
• Fall '19
• Mathématiques, Lycée Louis-le-Grand

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