Question

se tgx=2, entao o valor de cos 2x/1+sen 2x é?

Answer 1

sec²x = 1 + tg²x 
sec² x = 1 + 2²
sec² x = 1+4
sec²x = 5
sec x = √5

cosx = 1/secx 
cos x = 1/√5
cos x = √5/5

sen²x + cos²x = 1
sen²x + (√5/5)² = 1
sen²x + 1/5 = 1
sen²x = 1 - 1/5
sen²x = (5 - 1)/5
sen²x = 4/5
sen x= √4/5
senx = 2/√5
senx = 2√5/5

cos2x = cos²x - sen²x
cos 2x = (√5/5)² - (2√5/5)²
cos 2x = 1/5 - 4/5
cos2x = -3/5

sen2x = 2*senx*cosx
sen2x = 2*2√5/5*√5/5
sen2x = 20/25
sen2x = 4/5

Então:

cos2x/(1+sen2x) = (-3/5)/(1 +4/5) = (-3/5) / (9/5)
cos2x/(1+sen2x) = -3/5 * 5/9 = - 15/45 = - 1/3