Verified answer

1) sachant que tan (a) = - 1/2

Calculer tan (2 a) , cos (2 a) , sin (2 a)

tan (2a) = 2 tan(a)/(1 - tan² (a)) = 2 x (-1/2)/(1 - (- 1/2)²) = - 1/(1 - 1/4) = - 4/3

tan² (a) = (1 - cos (2a))/(1 + cos (2a)) = 1/4

⇔ 1 + cos (2 a) = 4(1 - cos (2a) ⇔ 1 + cos (2a) = 4 - 4 cos (2a)

⇒5 cos (2a) = 3 ⇒ cos (2a) = 3/5

tan (2a) = sin (2a)/cos (2a) = - 4/3 ⇒ sin (2a) = - 4/3 cos (2a) = - 4/3 x 3/5 = - 4/5

2) Ecrire en fonction de cos (2 x):

A = 5 cos² (x) - 3 sin² (x) = 5[(1 + cos(2x))/2] - 3 [(1 - cos (2x))/2]

⇔ 5 + 5 cos (2x) - 3 + 3 cos (2 x)]/2

⇔ 2 + 8 cos (2 x))/2

⇔ 1 + 4 cos (2 x)

A = 1 + 4 cos (2 x)

B = 2 cos² (x) + 3 sin² (x) - 4 sin² (x) *cos² (x)

= 2[(1 + cos (2 x))/2] + 3[(1 - cos (2 x))/2] - 4[(1 - cos (2 x))/2) * ((1 + cos (2 x))/2)

= 2 + cos (2 x) + 3 - 3 cos (2 x)]/2 - 4[(1 - cos² (2 x)]4

= 5 - 2 cos (2 x)]/2 - 1 + cos² (2 x)

= cos² (2 x) - cos (2 x) - 1 + 5/2

= cos² (2 x) - cos (2 x) +3/2

Vous faite le reste