https://file.aiquickdraw.com/mathbot/file/1725508569248xxa1fc9f.jpg

The image you provided contains the following mathematical expression:

\[
\cos^2 x + \cos^2 \left( x - \frac{\pi}{3} \right) - \cos^2 \left( x - \frac{\pi}{3} \right) = \frac{3}{2} - \frac{2}{3}
\]

Let's simplify and solve this equation step by step:

### Step 1: Simplify the Right-Hand Side
First, simplify the expression on the right-hand side:

\[
\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}
\]

So, the equation becomes:

\[
\cos^2 x + \cos^2 \left( x - \frac{\pi}{3} \right) - \cos^2 \left( x - \frac{\pi}{3} \right) = \frac{5}{6}
\]

### Step 2: Cancel Out Terms
Notice that \(\cos^2 \left( x - \frac{\pi}{3} \right)\) appears twice, once with a positive sign and once with a negative sign. Therefore, these terms cancel each other out, leaving:

\[
\cos^2 x = \frac{5}{6}
\]

### Step 3: Solve for \(x\)
Now we solve the equation:

\[
\cos^2 x = \frac{5}{6}
\]

Taking the square root of both sides:

\[
\cos x = \pm \sqrt{\frac{5}{6}}
\]

Thus, \(x\) can be expressed as:

\[
x = \cos^{-1}\left( \pm \sqrt{\frac{5}{6}} \right)
\]

This gives two possible solutions for \(x\), depending on the sign chosen.

Would you like more details on how to express the final solutions or explore the general form? Also, here are a few related questions:

1. What is the general solution to \(\cos x = \pm \sqrt{\frac{5}{6}}\)?
2. How can trigonometric identities be used to simplify similar expressions?
3. What other types of equations can we solve using cosine?
4. Can you apply this process to other trigonometric functions like sine or tangent?
5. What are the common values of cosine and sine for angles like \(\frac{\pi}{3}\)?

**Tip**: When dealing with trigonometric identities, always look for terms that can cancel each other out to simplify the equation quickly.

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

Simplify and solve the trigonometric equation involving cosine: cos^2(x) + cos^2(x - π/3) - cos^2(x - π/3) = 3/2 - 2/3. Explore trigonometric identities and solve for x with a detailed step-by-step solution. Suitable for high school students in Grades 10-12.

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