Understanding the Problem
To solve the expression \( \cos \theta = \frac{2}{3} \) and the equation \( 2 \sec^2 \theta + 2 \tan^2 \theta - 9 \), we will break it down step by step.

Step 1: Identify Secant and Tangent
- The relationship between secant and cosine is:
\( \sec \theta = \frac{1}{\cos \theta} \)
Therefore,
\( \sec^2 \theta = \frac{1}{\cos^2 \theta} \)
- The relationship between tangent and cosine is:
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
and
\( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \)
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \(\sin^2 \theta\) in terms of \(\cos^2 \theta\).

Step 2: Substitute Values
Given \( \cos \theta = \frac{2}{3} \), we can find \( \cos^2 \theta \):
- \( \cos^2 \theta = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)
Now, find \( \sec^2 \theta \) and \( \tan^2 \theta \):
- \( \sec^2 \theta = \frac{1}{\cos^2 \theta} = \frac{9}{4} \)
- \( \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9} \)
- \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\frac{5}{9}}{\frac{4}{9}} = \frac{5}{4} \)

Step 3: Substitute into the Equation
Now, substitute these values back into the equation:
\( 2 \sec^2 \theta + 2 \tan^2 \theta - 9 \)
- \( 2 \cdot \frac{9}{4} + 2 \cdot \frac{5}{4} - 9 \)
- \( \frac{18}{4} + \frac{10}{4} - 9 = \frac{28}{4} - 9 = 7 - 9 = -2 \)

Conclusion
The value of the expression \( 2 \sec^2 \theta + 2 \tan^2 \theta - 9 \) is \( -2 \). This step-by-step breakdown allows you to understand the relationships and substitutions used in trigonometric identities.