Question

prove
cos^2(A+120)+cos^2(A-120)+cos^2(A)=1.5

Answer 1

Answer:

The proof is as follows

Cos 2A = 2Cos2(A) – 1

Cos 2A + 1 = 2 Cos 2 ( A )  

(1 + Cos 2A)  /2 = Cos 2A

i.e  Cos 2A = (1 + Cos 2A)/2

\begin{gathered}{array}{l}{\frac{1+\cos (2(A-120))}{2}+\cos 2 A+\frac{1+\cos (2(A+120))}{2}}\\ {\frac{1+\cos ((2 A-240))+1+1+\cos ((2 A+240))}{2}+\cos 2 A}\\ {1+\frac{\cos ((2 A-240))+1+1+\cos ((2 A+240))}{2}+\cos 2 A}\\ {1+\frac{2 \cos 2 A \cos 240}{2}+\cos 2 A}{array}\end{gathered}arrayl21+cos(2(A−120))+cos2A+21+cos(2(A+120))21+cos((2A−240))+1+1+cos((2A+240))+cos2A1+2cos((2A−240))+1+1+cos((2A+240))+cos2A1+22cos2Acos240+cos2Aarray

1 + Cos2 A( -1/2  ) + Cos2 A

1- ( 2 Cos 2A -1) (  1/2  ) + Cos2A

1 – Cos2A + (  1/2  ) + Cos2 A

= 3/2

Answer 2

Answer

cos

2

A+cos

2

(120

∘

+A)+cos

2

(120

∘

−A)=cos

2

A+

2

1

(1+cos(240

∘

+2A))+

2

1

(1+cos(240

∘

−2A))

=cos

2

A+1+

2

1

(cos(240

∘

+2A)+cos(240

∘

−2A))

=1+cos

2

A+

2

1

(2cos240

∘

cos2A)

=1+cos

2

A−

2

1

(2cos

2

A−1)

=1+

2

1

=

2

3