Proof suf Re {m LH e^i2 pi tc } = m H cos (2 pi tc t) - ma LHsin (2 pi te f) L.H.S Re {MLH e I 2 pi te t} suppose mlH is a complex number MLH = M_1 L_H + I m_2 LH and e i2 pi tct = cos (2 pi tc t) + I sin in (2 pi tc t) rightarrow Re {(m1 LH + I m2 LH) cos 2 pi tc t + I sin (2 pi tc t)) rightarrow re {m2 LH cos 2 pi tc t - m LH sin 2 pi tc t) + I m_2 LH cos (2 pi tc t) + j^2 m_2 LH sin (2 pi tc t) rightarrow Re {m_1 LH cos (2 pi tc t) - mg(2 pi tc t) + j (m_1 LH sin 2 pi tc + m_1 LH cos 2 pi tc t) rightarrow Re {x + jy} = x {j^2 = -2 from eq^n 1 rightarrow m_1 LH cos (2 pi tc t) - m2 LH s in (2 pi tc t) = R.H.S