Maximum value of $f(x) = \cos x \left( \sin x + \sqrt {\sin^2x +\sin^2a}\right)$ Can we find maximum value of $$f(x) = \cos x \left(  \sin x + \sqrt
{\sin^2x +\sin^2a}\right)$$
where '$a$' is a given constant.
Using derivatives makes calculation too complicated.

the first derivative is given by $$f'(x)=-\sin \left( x \right)  \left( \sin \left( x \right) +\sqrt { \left( 
\sin \left( x \right)  \right) ^{2}+ \left( \sin \left( a \right) 
 \right) ^{2}} \right) +\cos \left( x \right)  \left( \cos \left( x
 \right) +{\frac {\sin \left( x \right) \cos \left( x \right) }{\sqrt 
{ \left( \sin \left( x \right)  \right) ^{2}+ \left( \sin \left( a
 \right)  \right) ^{2}}}} \right) 
$$ simplifying this you have to solve the equation for $x$:
$$- \left( \sin \left( x \right) +\sqrt { \left( \sin \left( x \right) 
 \right) ^{2}+ \left( \sin \left( a \right)  \right) ^{2}} \right) 
 \left( \sin \left( x \right) \sqrt { \left( \sin \left( x \right) 
 \right) ^{2}+ \left( \sin \left( a \right)  \right) ^{2}}- \left( 
\cos \left( x \right)  \right) ^{2} \right) 
=0$$ 
