Trigonometric Identities

We are given the equation:

We need to find the value of the expression .

By using the identity for the sum of inverse cosines:

Thus,

However, rather than solving this completely, we can simplify the given expression using the identity:

This identity suggests a trigonometric simplification because the original expression 9x² - 12xy cosθ + 4y² can also be seen in the context of its quadratic completion.

Observing the trigonometric identity and simplification, the expression:

Finally due to the structure of the equation and considering all simplifications, the appropriate simplification is

Hence, the value of the expression is .