Answer
\begin{aligned}
T_{n}(x) =\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{2}}\left(x-\frac{\pi}{4}\right)-\frac{1}{2 \sqrt{2}}\left(x-\frac{\pi}{4}\right)^{2}+\frac{1}{6 \sqrt{2}}\left(x-\frac{\pi}{4}\right)^{3}++\ldots \frac{1}{n ! \sqrt{2}}\left(x-\frac{\pi}{4}\right)^{n}
\end{aligned}
Work Step by Step
Since
\begin{array}{ll}
{f(x)=\cos x,} & {f(\pi/4)=\frac{\sqrt {2}}{2}} \\
{f^{\prime}(x)=-\sin x,} & {f^{\prime}(\pi/4)=\frac{-\sqrt {2}}{2}}\\
{f^{\prime \prime}(x)=-\cos x ,} & {f^{\prime \prime}(\pi/4)=\frac{-\sqrt {2}}{2}}\\
{f^{\prime \prime \prime}(x)=\sin x,} & {f^{\prime \prime \prime}(\pi/4)=\frac{\sqrt {2}}{2}}
\end{array}
Then
\begin{aligned}
T_{n}(x) &=\frac{\sqrt{2}}{2}+\frac{-\frac{\sqrt{2}}{2}}{1 !}\left(x-\frac{\pi}{4}\right)+\frac{-\frac{\sqrt{2}}{2}}{2 !}\left(x-\frac{\pi}{4}\right)^{2}+\frac{\frac{\sqrt{2}}{2}}{3 !}\left(x-\frac{\pi}{4}\right)^{3}+\frac{\frac{\sqrt{2}}{2}}{4 !}\left(x-\frac{\pi}{4}\right)^{4}+\ldots\\
&=\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{2}}\left(x-\frac{\pi}{4}\right)-\frac{1}{2 \sqrt{2}}\left(x-\frac{\pi}{4}\right)^{2}+\frac{1}{6 \sqrt{2}}\left(x-\frac{\pi}{4}\right)^{3}++\ldots \frac{1}{n ! \sqrt{2}}\left(x-\frac{\pi}{4}\right)^{n}
\end{aligned}
