Sigh Sine Cos Cosine

Thanks for the advice from Harsh Shrivastava , the solution is as follows:

By Cauchy-Schwarz inequality:

$\begin{aligned} \left(\sin x \cos x + \cos x \sqrt{\sin^2 x +\frac 34}\right)^2 & \le \left(\sin^2 x + \cos^2 x \right) \left(\cos^2 x + \sin^2 x + \frac 34 \right) = (1)\left(1+\frac 34\right) = \frac 74 \\ \implies \cos x \left(\sin x + \sqrt{\sin^2 x +\frac 34}\right) & \le \frac {\sqrt 7}2 \approx \boxed{1.323} \end{aligned}$

Hint : When all fails, Cauchy Schwartz hails~!

(Though we can solve this problem without Cauchy Schwartz also but that would be too lengthy/)