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The second term can be expanded to $$ 2*|a_k|\left(\frac{ e^{i(2\pi*f_k*t+\angle a_k)} + e^{-i(2\pi*f_k*t+\angle a_k)}}{2}\right) $$

$$= |a_k|e^{i\angle a_k}e^{i(2\pi*f_k*t)}+|a_k|e^{-i\angle a_k}e^{-i(2\pi*f_k*t)}$$

$$= a_k e^{i(2\pi*f_k*t)}+a_k^*e^{-i(2\pi*f_k*t)}$$

Since $a_k=|a_k|e^{i\angle a_k}$ and $a_k^*=|a_k|e^{-i\angle a_k}$ and $cos\theta=\frac{ e^{i\theta} + e^{-i\theta}}{2}$

The second term can be expanded to $$\begin{align} 2|a_k|\left(\frac{ e^{i(2\pi f_k t+\angle a_k)} + e^{-i(2\pi f_k t+\angle a_k)}}{2}\right) &= |a_k|e^{i\angle a_k}e^{i(2\pi f_k t)}+|a_k|e^{-i\angle a_k}e^{-i(2\pi f_k t)} \\ &= a_k e^{i(2\pi f_k t)}+a_k^*e^{-i(2\pi f_k t)} \\ \end{align}$$

Since $a_k=|a_k|e^{i\angle a_k}$ and $a_k^*=|a_k|e^{-i\angle a_k}$ and $\cos\theta=\frac{ e^{i\theta} + e^{-i\theta}}{2}$

The second term can be expanded to $$ 2*|a_k|(\frac{ e^{i(2\pi*f_k*t+\angle a_k)} + e^{-i(2\pi*f_k*t+\angle a_k)}}{2}) $$

$$= |a_k|e^{i\angle a_k}e^{i(2\pi*f_k*t)}+|a_k|e^{-i\angle a_k}e^{-i(2\pi*f_k*t)}$$

$$= a_ke^{i\angle a_k}e^{i(2\pi*f_k*t)}+a_k^*e^{-i\angle a_k}e^{-i(2\pi*f_k*t)}$$

Since $a_k=|a_k|e^{i\angle a_k}$ and $a_k^*=|a_k|e^{-i\angle a_k}$ and $cos\theta=\frac{ e^{i\theta} + e^{-i\theta}}{2}$

The second term can be expanded to $$ 2*|a_k|\left(\frac{ e^{i(2\pi*f_k*t+\angle a_k)} + e^{-i(2\pi*f_k*t+\angle a_k)}}{2}\right) $$

$$= |a_k|e^{i\angle a_k}e^{i(2\pi*f_k*t)}+|a_k|e^{-i\angle a_k}e^{-i(2\pi*f_k*t)}$$

$$= a_k e^{i(2\pi*f_k*t)}+a_k^*e^{-i(2\pi*f_k*t)}$$

Since $a_k=|a_k|e^{i\angle a_k}$ and $a_k^*=|a_k|e^{-i\angle a_k}$ and $cos\theta=\frac{ e^{i\theta} + e^{-i\theta}}{2}$

The second term can be expanded to $$ 2*|a_k|(\frac{ e^{i(2\pi*f_k*t+a_k)} + e^{-i(2\pi*f_k*t+a_k)}}{2}) $$

$$= |a_k|e^{ia_k}e^{i(2\pi*f_k*t)}+|a_k|e^{-ia_k}e^{-i(2\pi*f_k*t)}$$

$$= a_ke^{ia_k}e^{i(2\pi*f_k*t)}+a_k^*e^{-ia_k}e^{-i(2\pi*f_k*t)}$$

Since $a_k=|a_k|e^{ia_k}$ and $a_k^*=|a_k|e^{-ia_k}$ and $cos\theta=\frac{ e^{i\theta} + e^{-i\theta}}{2}$

The second term can be expanded to $$ 2*|a_k|(\frac{ e^{i(2\pi*f_k*t+\angle a_k)} + e^{-i(2\pi*f_k*t+\angle a_k)}}{2}) $$

$$= |a_k|e^{i\angle a_k}e^{i(2\pi*f_k*t)}+|a_k|e^{-i\angle a_k}e^{-i(2\pi*f_k*t)}$$

$$= a_ke^{i\angle a_k}e^{i(2\pi*f_k*t)}+a_k^*e^{-i\angle a_k}e^{-i(2\pi*f_k*t)}$$

Since $a_k=|a_k|e^{i\angle a_k}$ and $a_k^*=|a_k|e^{-i\angle a_k}$ and $cos\theta=\frac{ e^{i\theta} + e^{-i\theta}}{2}$