Derivative of $$$1 - \frac{\sin{\left(t \right)}}{2}$$$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dt} \left(1 - \frac{\sin{\left(t \right)}}{2}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(1\right) - \frac{d}{dt} \left(\frac{\sin{\left(t \right)}}{2}\right)\right)}$$

Apply the constant multiple rule $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ with $$$c = \frac{1}{2}$$$ and $$$f{\left(t \right)} = \sin{\left(t \right)}$$$:

$$- {\color{red}\left(\frac{d}{dt} \left(\frac{\sin{\left(t \right)}}{2}\right)\right)} + \frac{d}{dt} \left(1\right) = - {\color{red}\left(\frac{\frac{d}{dt} \left(\sin{\left(t \right)}\right)}{2}\right)} + \frac{d}{dt} \left(1\right)$$

The derivative of the sine is $$$\frac{d}{dt} \left(\sin{\left(t \right)}\right) = \cos{\left(t \right)}$$$:

$$- \frac{{\color{red}\left(\frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}}{2} + \frac{d}{dt} \left(1\right) = - \frac{{\color{red}\left(\cos{\left(t \right)}\right)}}{2} + \frac{d}{dt} \left(1\right)$$

The derivative of a constant is $$$0$$$:

$$- \frac{\cos{\left(t \right)}}{2} + {\color{red}\left(\frac{d}{dt} \left(1\right)\right)} = - \frac{\cos{\left(t \right)}}{2} + {\color{red}\left(0\right)}$$

Thus, $$$\frac{d}{dt} \left(1 - \frac{\sin{\left(t \right)}}{2}\right) = - \frac{\cos{\left(t \right)}}{2}$$$.