Proving that cos^2x cos^2(x π/3) cos^2(x-π/3) = 3/2

To prove this trigonometric identity, we will use various trigonometric identities and algebraic manipulations. Let's break down the proof into smaller steps:



The first step is to establish the value of cos(π/3), which is a well-known identity. We can use the unit circle or the special triangle to find this value. Using the unit circle, we observe that at an angle of π/3, the x-coordinate of the point on the unit circle is 1/2. Therefore, cos(π/3) = 1/2.



The next step is to use the angle subtraction formula to express cos(x-π/3) in terms of cos(x) and sin(x). The angle subtraction formula states that cos(a-b) = cos(a)cos(b) + sin(a)sin(b). By substituting a = x and b = π/3, we get cos(x-π/3) = cos(x)cos(π/3) + sin(x)sin(π/3).



Now, let's substitute the values from Step 1 and Step 2 into the given expression and simplify it:

cos^2x cos^2(x π/3) cos^2(x-π/3) = cos^2x cos^2(x π/3) (cos(x)cos(π/3) + sin(x)sin(π/3))^2

Expanding the square of the expression inside the parentheses:

= cos^2x cos^2(x π/3) (cos^2(x)cos^2(π/3) + 2cos(x)sin(x)cos(π/3)sin(π/3) + sin^2(x)sin^2(π/3))

Using the identity sin^2(π/3) = 3/4 and cos^2(π/3) = 1/4:

= cos^2x cos^2(x π/3) (cos^2(x)/4 + 2cos(x)sin(x)/4 + 3sin^2(x)/4)

Multiplying everything by 4 to clear the denominators:

= cos^2x cos^2(x π/3) (cos^2(x) + 2cos(x)sin(x) + 3sin^2(x))



Using the Pythagorean identity, cos^2(x) + sin^2(x) = 1, we can simplify the expression further:

= cos^2x cos^2(x π/3) (1 + cos(x)sin(x) + 2sin^2(x))