Answer to Question #177371 in Trigonometry for Yusuf Idris

Question

Show by expanding (cosa + isina) (cosB + isinB) and using trigonometry Identities that (cosa + isina)(cosB +isinB)= cos(a+B) + isin(a+B)

Answer 1

Given, a trigonometric expression (cosa + isina) (cosB + isinB).

(cosa + isina) (cosB + isinB)= cosa (cosB + isinB) + isina(cosB + isinB)

=cosa cosB + icosa sinB + isina cosB - sina sinB

Separate real and imaginary part.

=cosa cosB - sina sinB + i(sina cosB + cosa sinB)

Trigonometry indenties, cos(x+y)=cosx cosy - sinx siny

and sin(x+y)=sinx cosy + cosx siny

Therefore,

(cosa + isina) (cosB + isinB) = cos(a+B) + isin(a+B)