Confusion regarding use of inverse Euler formula?

Question

As shown underlined in attached photo

How do we get last line? I know how we got 1/2 term, but I am confused how we got the terms $\cos(11\pi*t -\pi/2)$ and $\cos(9\pi*t-\pi/2).$

Answer 1

Hint:

$$\Large e^{-j\pi/2}=-e^{j\pi/2}$$

Answer 2

Alternatively, instead of multiplying the sine by $-j=e^{-j\pi/2}$ to get the second line, you can express the sine as a cosine:

$$\begin {aligned} x(t) &= \cos (\pi t) \sin (10 \pi t) \\ &= \cos (\pi t) \cos (10 \pi t - \pi / 2) \\ &= \frac {1} {4} \left( e^{j \pi t} + e^{-j \pi t} \right) \left( e^{j (10 \pi t - \pi /2)} + e^{-j (10 \pi t - \pi / 2)} \right) \\ &= \frac{1} {4} \left[ e^{j (11 \pi t - \pi / 2)} + e^{-j (11 \pi t - \pi / 2)} + e^{j(9 \pi t - \pi /2)} + e^{-j(9 \pi t - \pi /2)} \right] \\ &= \frac {1} {2} \cos (11 \pi t - \pi / 2) + \frac {1} {2} \cos(9 \pi t - \pi / 2) \end{aligned}$$