QPaperGen

We evaluate each term:

\begin{align*}
\sin\left(\frac{3\pi}{4}\right) &= \sin\left(\pi - \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \\
\Rightarrow \sin^2\left(\frac{3\pi}{4}\right) &= \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \\
\\
\cos\left(\frac{\pi}{4}\right) &= \frac{1}{\sqrt{2}} \\
\Rightarrow \cos^2\left(\frac{\pi}{4}\right) &= \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \\
\\
\sec\left(\frac{\pi}{3}\right) &= \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2 \\
\Rightarrow \sec^2\left(\frac{\pi}{3}\right) &= 2^2 = 4
\end{align*}

Now substitute into the original expression:

\begin{align*}
2\sin^2\left(\frac{3\pi}{4}\right) + 2\cos^2\left(\frac{\pi}{4}\right) + 2\sec^2\left(\frac{\pi}{3}\right)
&= 2 \cdot \frac{1}{2} + 2 \cdot \frac{1}{2} + 2 \cdot 4 \\
&= 1 + 1 + 8 \\
&= 10
\end{align*}

Hence proved.