# cosx + cosy=3/2 and sinx + siny=1/2 and theta is the A.M. of x and y then sin(2theta)+cos(2theta)=

arun
123 Points
7 years ago
$\cos x + \cos y = \frac{3}{2}$
$2\cos (\frac{x + y}{2})\cos (\frac{x-y}{2}) = \frac{3}{2}$             $\left [\because \cos x + \cos y = 2\cos (\frac{x + y}{2})\cos (\frac{x-y}{2}) \right ]$
and $\theta = \frac{x + y}{2}$
so, $2\cos (\theta )\cos (\frac{x-y}{2}) = \frac{3}{2}$                                       …................................................1
similarly, $2\sin (\theta )\cos (\frac{x-y}{2}) = \frac{1}{2}$                              ….................................................2
dividing equation 2 by 1
$\tan\theta = \frac{1}{3}$
so, $\sin\theta = \frac{1}{\sqrt{10}}$
and $\cos\theta = \frac{3}{\sqrt{10}}$
now using the formula for sin2theta and cos2theta i.e
$\sin 2\theta + \cos 2\theta = 2\sin \theta\cos \theta + \cos^{2} \theta - \sin^{2} \theta$
$\sin 2\theta + \cos 2\theta = \frac{3}{5} + \frac{4}{5}$
$\sin 2\theta + \cos 2\theta = \frac{7}{5}$