Answer
$R_{n}(x)=\dfrac{\cos [\dfrac{(n+1)\pi}{2}+c](x-\dfrac{\pi}{2})^{(n+1)}}{(n+1)!}$
Work Step by Step
The Remainder Theorem for some points $c$ and $a$ can be expressed as:
$R_{n}(x)=\dfrac{f^{n+1}(c)(x-a)^{n+1}}{(n+1)!}~~~~~(1)$
We are given the functions $f(x)=\cos x$ and $a=\dfrac{\pi}{2}$
Put these values in the equation (1) to obtain:
$R_{n}(x)=\dfrac{(-1)^{(n+1)}(e^{-c})(x-0)^{(n+1)}}{(n+1)!}$
Thus, we have: $R_{n}(x)=\dfrac{\cos [\dfrac{(n+1)\pi}{2}+c](x-\dfrac{\pi}{2})^{(n+1)}}{(n+1)!}$
