Question

Prove that $(\cos x+\cos y{)}^2+(\sin x-\sin y{)}^2=4{\cos }^2\left(\frac{x+y}{2}\right)$

Answer 1

Given that:

$LHS=(cosx+cosy{)}^2+(sinx-siny{)}^2$

$=co{s}^{2}x+co{s}^{2}y+2cosxcosy+si{n}^{2}x+si{n}^{2}y-2sinxsiny$

$=1+1+2(cosxcosy-sinxsiny)$

$=2[1+cos(x+y\left)\right]$

$=2[1+2co{s}^2\left(\frac{x+y}{2}\right)-1]$ [$cos\theta =2co{s}^2\frac{\theta }{2}-1$]

$=4co{s}^2\left(\frac{x+y}{2}\right)$

$=RHS$

Hence proved.