The period of $\cos x\cos \left(120°-x\right)\cos \left(120°+x\right)$  is

To determine the period of the given expression cos x cos(120° − x) cos(120° + x), we begin by simplifying it step-by-step.

Simplification

The expression can be written as: 
cos x cos(120° − x) cos(120° + x). 
Using the product-to-sum formula for cos(A)cos(B), we simplify:

cos x cos(120° − x) cos(120° + x) becomes: 
1/2 . cos x . 2 cos(120° − x) cos(120° + x).

Using the identity 2 cos A cos B = cos(A + B) + cos(A − B), we get: 
1/2 cos x [cos((120° − x) + (120° + x)) − cos((120° − x) − (120° + x))].

Simplify the terms inside the cosine functions:

• (120° − x) + (120° + x) = 240°
• (120° − x) − (120° + x) = −2x

So, the expression becomes: 
1/2 cos x [cos(240°) − cos(−2x)].

Using the fact that cos(−θ) = cos(θ) and cos(240°) = −cos(120°), we rewrite: 
1/2 cos x [−cos(120°) − cos(2x)].

Expanding further: 
−1/2 cos x . cos(120°) − 1/2 cos x . cos(2x).

Using the product-to-sum formula again for cos x . cos(2x): 
−1/4 cos x − 1/4 [cos(x + 2x) − cos(x − 2x)].

Simplify the terms: 
−1/4 cos x − 1/4 cos(3x) + 1/4 cos x.

Combining like terms: 
−1/4 cos(3x).

Period Calculation

The final simplified expression is proportional to cos(3x). The period of cos(3x) is determined by the formula: 
Period = 2π / 3.

Final Answer

Therefore, the period of cos x cos(120° − x) cos(120° + x) is: 
2π / 3.

Note:

The value of cos 120 degrees plays a critical role in the simplification as it equals −1/2, which affects the overall expression. The identity for cos 120 degrees is used repeatedly to ensure accurate simplifications.