91Ó°ÊÓ

Trigonometric identities are fundamental tools in trigonometry. They are equations involving trigonometric functions that hold true for all values of the variable angles. Some commonly used trigonometric identities include the Pythagorean identity: \[\sin^2 \theta + \cos^2 \theta = 1\] and angle addition formulas: \[\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\] and \[\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b.\] These identities simplify complex trigonometric problems and serve as foundational elements for solving equations or transforming expressions.
In the given problem, recognizing that \[\cos^2 \gamma = (\cos \gamma)^2\] helped us match the structure to a quadratic equation, making it easier to factorize.

Quadratic equations are polynomial equations of the form \[ax^2 + bx + c = 0.\] These equations can be factored, solved using the quadratic formula, or completed in the square.
In the original exercise, we identified a quadratic equation format within our trigonometric expression: \[\cos^2 \gamma - \cos \gamma - 6.\] By substituting \[x = \cos \gamma,\] the problem became a standard quadratic: \[x^2 - x - 6.\]
The quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] is handy, and so is the factorization method, where we look for two numbers that multiply to \[ac\] and add up to \[b.\]

Factoring is a valuable algebraic technique used to simplify expressions or solve equations. Common factoring methods include finding the greatest common factor, difference of squares, and grouping.
In this case, we utilized factoring of quadratic expressions. We aimed to express the quadratic \[x^2 - x - 6\] in the form \[(x - p)(x - q).\] Here, \[p\] and \[q\] are roots of the equation, i.e., \[p \times q = ac\] and \[p + q = b.\] For \[x^2 - x - 6,\] the pairs that work are \[-3\] and \[+2.\] Thus, we wrote it as: \[(x - 3)(x + 2).\]
Finally, we substituted back \[x = \cos \gamma\] to derive the factored trigonometric expression: \[(\cos \gamma - 3) (\cos \gamma + 2).\]