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left side=Show that :
\(\displaystyle \left( {\sin x + a\cos x} \right)\left( {\sin x + b\cos x} \right) \leq 1 + \left( \frac{a + b}{2} \right)^2\)
Yes, it is correct...it appears you are assuming the two sinusoidal factors are in phase with one another, that is for $a=b$. In this case, then your result is equivalent to that which I gave.left side=
\(\displaystyle \left( {\sin x + a\cos x} \right)\left( {\sin x + b\cos x} \right)\leq \sqrt{1+a^2}\times \sqrt{1+b^2}\)
$\leq\dfrac{1+a^2+1+b^2}{2}=1+\dfrac {a^2+b^2}{2}$
Are you sure , right side is correct ?
It appears you are on the right track here, but have made some algebraic errors.if it is correct then ,we must prove
$\dfrac {a^2+b^2}{2}\leq (\dfrac{a+b}{2})^2=\dfrac {a^2+b^2}{4}+{ab}$
for all $a,b \in R$
${\therefore \dfrac {a^2+b^2}{4}\leq ab}$
how about if ab<0,then it does not fit
Your errors are quite minor, and in fact leads to a much simpler approach than I have.sorry ,I have made some algebraic errors
I will try to use another approach