Question

demonstrati egalitatea: sin(a+b)sin(a-b)=sin^2a-sin^2b

Answer 1

sin(a+b)=sina cosb+sinbcosa
sin(a-b)=sina cosb-sinb cos a
sin(a+b)sin(a-b)=(sina cosb+sinbcosa)(sina cosb-sinb cos a)
=[tex] (sinacosb)^{2}-(sinbcosa)^{2}= \\ =sin^{2}a cos^{2}b- sin^{2}b cos^{2}a= \\ = sin^{2}a(1-sin^{2}b)-sin^{2}b(1-sin^{2}a)= \\ = sin^{2}a- sin^{2}a sin^{2}b- sin^{2}b+ sin^{2}b sin^{2}a= \\ = sin^{2}a- sin^{2} b. [/tex]