Displacement vector,
$\begin{aligned}
\mathbf{s} & =x(\hat{\mathbf{i}})+y(\hat{\mathbf{j}}) \\
& =a \sin (\omega t)(\hat{\mathbf{i}})+a[1-\cos (\omega t)](\hat{\mathbf{j}})
\end{aligned}$
Velocity vector,
$\begin{aligned}
\mathbf{v} & =\frac{d \mathbf{s}}{d t}=\frac{d}{d t}[a \sin (\omega t)(\hat{\mathbf{i}})+a\{1-\cos (\omega t)\}(\hat{\mathbf{j}})] \\
& =a \omega \cos (\omega t)(\hat{\mathbf{i}})+a[0-\{-\omega \sin (\omega t)\}](\hat{\mathbf{j}}) \\
& =a \omega \cos (\omega t)(\hat{\mathbf{i}})+a \omega \sin (\omega t)(\hat{\mathbf{j}})
\end{aligned}$
Acceleration vector,
$\begin{aligned}
\mathbf{a}=\frac{d \mathbf{v}}{d t} & =\frac{d}{d t}[a \omega \cos (\omega t)(\hat{\mathbf{i}})+a \omega \sin (\omega t)(\hat{\mathbf{j}})] \\
& =-a \omega^2 \sin (\omega t)(\hat{\mathbf{i}})+a \omega^2 \cos (\omega t)(\hat{\mathbf{j}})
\end{aligned}$
Now, calculation of angle between $\mathbf{v}$ and $\mathbf{a}$
$\cos \theta=\frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}||\mathbf{a}|}$
$[a \omega \cos (\omega t)(\hat{\mathbf{i}})+a \omega \sin (\omega t)(\hat{\mathbf{j}})]$
$\begin{array}{r}
\cos \theta=\frac{\cdot\left[-a \omega^2 \sin (\omega t)(\hat{\mathbf{i}})+a \omega^2 \cos (\omega t)(\hat{\mathbf{j}})\right]}{\sqrt{[a \omega \cos (\omega t)]^2+[a \omega \sin (\omega t)]^2}} \\
\times \sqrt{\left[-a \omega^2 \sin (\omega t)\right]^2+\left[a \omega^2 \cos (\omega t)\right]^2}
\end{array}$
$\begin{gathered}\cos \theta=\frac{-a^2 \omega^3 \cos (\omega t) \sin (\omega t)+a^2 \omega^3 \sin (\omega t) \cos (\omega t)}{\sqrt{a^2 \omega^2 \cos ^2(\omega t)+a^2 \omega^2 \sin ^2(\omega t)}} \\ \times \sqrt{a^2 \omega^4 \sin ^2(\omega t)+a^2 \omega^4 \cos ^2(\omega t)} \\ \cos \theta=\frac{0}{\sqrt{a^2 \omega^2 \cos ^2(\omega t)+a^2 \omega^2 \sin ^2(\omega t)}} \\ \times \sqrt{a^2 \omega^4 \sin ^2(\omega t)+a^2 \omega^4 \cos ^2(\omega t)} \\ \cos \theta=0 \Rightarrow \theta=\frac{\pi}{2}\end{gathered}$