Question

Wykaż tożsamość:

Answer 1

 V[(1-cosx)/(1+cosx)] + V[(1+cosx)/(1-cosx)] = 2/IsinxI  I^2  (obie strony podnoszę do 2-giej) 

{V[(1-cosx)/(1+cosx)] + V[(1+cosx)/(1-cosx)]}^2 = (2/IsinxI)^2

L^2 = (1-cosx)/(1+cosx) + 2V[(1-cosx)(1+cosx)/(1+cosx)(1-cosx)] + (1+cosx)/(1-cosx) = 

  = [(1-cosx)(1-cosx) + (1+cosx)(1+cosx)]/[(1+cosx)(1-cosx)] +2V[(1-cos^2(x))/(1-cos^2(x))] = 

  = (1 - cosx - cosx + cos^2(x) + 1 + cosx + cosx + cos^2(x))/sin^2(x) + 2*1 = 

  = [(2+2cos^2(x) + 2(1-cos^2(x))]/sin^2(x) = (2 + 2cos^2(x) + 2 - 2cos^2(x))/sin^2(x) = 4/sin^2(x) 

P^2 = 4/sin^2(x)

L = V(4/sin^2(x)) = 2/IsinxI 

P = V(4/sin^2(x)) = 2/IsinxI 

L = P 

 

 

Oznaczenia: 

V  -  pierwiastek kwadratowy 

sin^2(x)  -  sinus kwadrat x 

cos^2(x)  - cosinus kwadrat x