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There is a mistake in the last line:
~~$$\eqalign{V_{C_1}(t) &= V_1(t) - RI_{C_1}(t) \cr &= \sin(t) - (\sin(t) + \cos(t) - e^{-t})/2 \cr~~
~~&= (\sin(t) - \cos(t) + e^{-t})/2 \cr~~
~~&= (\cos(t-90\ deg) - (-\sin(t-90\ deg)) + e^{-t})/2\cr~~
~~&= (\cos(t-90\ deg) + \sin(t-90\ deg) + e^{-t})/2\cr~~
~~&= I_{C_1}(t - 90 \ deg)}.$$~~

There is a mistake in the last line:
$$\eqalign{V_{C_1}(t) &= V_1(t) - RI_{C_1}(t) \cr &= \sin(t) - \dfrac{\sin(t) + \cos(t) - \mathrm{e}^{-t}}{2} \cr
&= \dfrac{\sin(t) - \cos(t) + \mathrm{e}^{-t}}{2} \cr
&= \dfrac{\cos(t-90^\circ) - [-\sin(t-90^\circ)] + \mathrm{e}^{-t}}{2}\cr
&= \dfrac{\cos(t-90^\circ) + \sin(t-90^\circ) + \mathrm{e}^{-t}}{2}\cr
&= I_{C_1}(t - 90^\circ)}.$$