Question

Show that: $\cos 24^{\circ}+\cos 55^{\circ}+\cos 125^{\circ}+\cos 204^{\circ}+\cos 300^{\circ}=\frac{1}{2}$

Answer 1

L.H.S. = cos24+cos204+cos55 + cos125 +cos300  

= cos 24 + cos (180+24) + sin (90-55) + cos (90+35) + cos (360-60) [ cos x = sin (90-x)]  

= cos 24 - cos 24 + sin 35 - sin 35 + cos 60 [ cos (180 + x ) = - cosx ; cos (90 + y ) = - sin y ; cos (360 - x ) = cos x ]  

= (cos 24 - cos 24) + (sin 35 - sin 35 )+ cos 60  

= 0 + 0 + 1/2 = 1/2 = R.H.S.