Question

prove that cos² x + cos²(x+π/3) + cos²(x-π/3) = 3/2​

Answer 1

Answer:

LHS=cos

2

x+cos

2

(x+

3

π

)+cos

2

(x−

3

π

)

Weknowthat,cos2x=2cos

2

x−1

∴cos

2

x=

2

cos2x+1

∴cos

2

(x+

3

π

)=

2

cos2(x+

3

π

)+1

∴cos

2

(x−

3

π

)=

2

cos2(x−

3

π

)+1

∴LHS=1+

2

cos2x

+

2

1+cos(2x+

3

2π

)

+

2

1+cos(2x−

3

2π

)

=

2

1

[1+1+1+cos2x+cos(2x+

3

2π

)+cos(2x−

3

2π

)]

=

2

1

[3+cos2x+2cos

⎝

⎜

⎜

⎜

⎛

2

2x+2x+

3

2π

−

3

2π

⎠

⎟

⎟

⎟

⎞

].

⎝

⎜

⎜

⎜

⎛

cos

⎝

⎜

⎜

⎜

⎛

2

2x−2x+

3

2π

+

3

2π

⎠

⎟

⎟

⎟

⎞

⎠

⎟

⎟

⎟

⎞

=

2

1

[3+cos2x+2cos2x.cos

3

2π

]

=

2

1

[3+cos2x+2cos2x(−cos

3

π

)]

=

2

1

[3+cos2x−cosx]

=

2

3

=RHS

Answer 2

hope it's helpful........