Question

Demonstrati ca (1+sin2x)/(1+cos2x)=1/2(1+tgx)^2

Pentru orice numar real x

Answer 1

(1+sin2x)/(1+cos2x) =

= (sin²x+cos²x+2sinxcosx)/

(sin²x+cos²x+cos²x-sin²x) =

= (sinx+cosx)²/(2cos²x) =

= (1/2)•(sinx+cosx)²/(cosx)² =

= (1/2)•[(sinx+cosx)/(cosx)]² =

= (1/2)•(sinx/cosx + cosx/cosx)² =

= (1/2)•(tgx + 1)² =

= (1/2)•(1 + tgx)²