Answer
\[f(x)=\cos x\;,\;a=\frac{π}{6}\]
Work Step by Step
It is given that \[2\int_{a}^{x}f(t)\:dt=2\sin x-1\;\;\;...(1)\]
We will use the result \[\frac{d}{dx}\int_{f(x)}^{g(x)}F(t)\:dt=F(g(x))\cdot g'(x)-F(f(x))\cdot f'(x)\;\;\;...(2)\]
Differentiating (1) with respect to $x$ using (2)
\[\Rightarrow 2[f(x)\cdot (x)'-f(a)\cdot (a)'=2\cos x\]
\[\Rightarrow 2f(x)=2\cos x\]
\[\Rightarrow f(x)=\cos x\]
From (1)
\[2\int_{a}^{x}\cos t\:dt=2\sin x-1\]
\[2[\sin x-\sin a]=2\sin x-1\]
\[-1=-2\sin a\]
\[\sin a=\frac{1}{2}\]
\[a=\frac{π}{6}\]
Hence , \[f(x)=\cos x\;,\;a=\frac{π}{6}\]
