Question

Stiind ca x∈(0,\pi /2) si tgx=1/2 ,calculati : cos2x/(1+sin2x)

Answer 1

tg x=1/2
sinx/cosx=1/2
Ridicand la patrat obtinem:
sin²x/cos²x=1/4
Stim ca sin²x=1-cos²x si obtinem:
(1-cos²x)/cos²x=1/4
1/cos²x-cos²x/cos²x=1/4
1/cos²x-1=1/4
1/cos²x=1/4+1
1/cos²x=5/4
cos²x=4/5
x∈(0;π/2) => cosinusul are valoare pozitiva.
cos x=√4/5=2/√5=2√5/5
sin²x=1-cos²x
sin²x=1-4/5
sin²x=1/5
x∈(0;π/2) => sinusul are valoare pozitiva.
sin x=1/√5
sin x=√5/5

cos 2x=cos²x-sin²x=4/5-1/5=3/5
sin 2x=2sinxcosx=2*2√5/5*√5/5=2*2√5*√5/5*5=20/25=4/5

cos2x/(1+sin 2x)=3/5/4/5=3/5*5/4=3/4