Prove the following:cos2 x + cos2 (x + 120°) + cos2 (x – 120°) = 3

1.	Prove the following:cos2 x + cos2 (x + 120°) + cos2 (x – 120°) = 3/2
Answer» cos² x + cos² (x + 120°) + cos² (x – 120°) = \(\frac{1+cos2x}{2}+\frac{1+cos2(X+120°)}{2} + \frac{1+cos 2(x-120°)}{2}\) ...[∵cos²θ = \(\frac{1+cos \,2θ}{2}\) \(\frac{3}{2}+\frac{1}{2}\)[cos 2x + cos(2x + 240°) + cos(2x 240°)] = \(\frac{3}{2}+\frac{1}{2}\)(cos 2x + cos 2x cos 240°— sin 2x sin 240° + cos 2x cos 240° + sin 2x sin 240°) =\(\frac{3}{2}+\frac{1}{2}\)(cos 2x + 2 cos 2x cos 240°) =\(\frac{3}{2}+\frac{1}{2}\)[cos 2x + 2 cos 2x cos( 180° + 60°)] = \(\frac{3}{2}+\frac{1}{2}\)[cos 2x + 2cos 2x(-cos 600)] = \(\frac{3}{2}+\frac{1}{2}\)[cos 2x —2 cos 2x(\(\frac{1}{2}\))] =\(\frac{3}{2}+\frac{1}{2}\)( cos 2x – cos 2x) = \(\frac{3}{2}+\frac{1}{2}\)(0) = \(\frac{3}{2}\)= R.H.S

Prove the following:cos2 x + cos2 (x + 120°) + cos2 (x – 120°) = 3/2

Answer»

cos² x + cos² (x + 120°) + cos² (x – 120°)

= \(\frac{1+cos2x}{2}+\frac{1+cos2(X+120°)}{2} + \frac{1+cos 2(x-120°)}{2}\)

...[∵cos²θ = \(\frac{1+cos \,2θ}{2}\)

\(\frac{3}{2}+\frac{1}{2}\)[cos 2x + cos(2x + 240°) + cos(2x 240°)]

= \(\frac{3}{2}+\frac{1}{2}\)(cos 2x + cos 2x cos 240°— sin 2x sin 240° + cos 2x cos 240° + sin 2x sin 240°)

=\(\frac{3}{2}+\frac{1}{2}\)(cos 2x + 2 cos 2x cos 240°)

=\(\frac{3}{2}+\frac{1}{2}\)[cos 2x + 2 cos 2x cos( 180° + 60°)]

= \(\frac{3}{2}+\frac{1}{2}\)[cos 2x + 2cos 2x(-cos 600)]

= \(\frac{3}{2}+\frac{1}{2}\)[cos 2x —2 cos 2x(\(\frac{1}{2}\))]

=\(\frac{3}{2}+\frac{1}{2}\)( cos 2x – cos 2x)

= \(\frac{3}{2}+\frac{1}{2}\)(0)

= \(\frac{3}{2}\)= R.H.S