Show that ​ (i) {(3+2i)/(2-3i)+(3-2i)/(2+3i)} is purely real, (ii) {(√7+i√3)/(√7- i√3)+(√7- i√3)/(√7+i√3)} is purely real.

Question

Show that

(i) \(\left\{\frac{(3+2i)}{(2-3i)}+\frac{(3-2i)}{(2+3i)}\right\}\) is purely real,

{(3+2i)/(2-3i)+(3-2i)/(2+3i)}

(ii) \(\left\{\frac{(\sqrt7+i\sqrt3)}{(\sqrt7-i\sqrt3)}+\frac{(\sqrt7-i\sqrt3)}{(\sqrt7+i\sqrt3)}\right\}\) is purely real.

{(√7+i√3)/(√7- i√3)+(√7- i√3)/(√7+i√3)}

Answer 1

Hence, the given equation is purely real as there is no imaginary part.

(ii) Given : \(\left\{\frac{(\sqrt7+i\sqrt3)}{(\sqrt7-i\sqrt3)}+\frac{(\sqrt7-i\sqrt3)}{(\sqrt7+i\sqrt3)}\right\}\)

Taking the L.C.M, we get

Hence, the given equation is purely real as there is no imaginary part.