Question

prove that :
$(\sin \theta +\cos \theta {)}^2+(\cos \theta +\sin \theta {)}^2\le 4$

Answer 1

Prove that,

${(\sin \theta +\cos \theta )}^2+{(\cos \theta +\sin \theta )}^2\ge 1$

L.H.S.

${(\sin \theta +\cos \theta )}^2+{(\cos \theta +\sin \theta )}^2$

$=2{(\sin \theta +\cos \theta )}^2$

$=2({\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta )$

$=2(1+\sin 2\theta )$

We know that

Maximum value of $\sin 2\theta =1$

So,

$=2(1+1)$

$=2\times 2$

$=4\ge 1$

Hence proved.