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The double angle formula is a crucial trigonometric identity that helps us rewrite expressions involving angles in simpler forms. For cosine, the formula is: \[ \cos(2x) = 2 \cos^2(x) - 1 \]. This means that the cosine of twice an angle equals twice the square of the cosine of the original angle minus one. In the problem you're working on, we used the version of the double angle formula as: \[ \cos(2x) = 2\cos^2(x) - 1 \]. This helps to transform and simplify trigonometric equations, providing an easier path to finding solutions. Understanding how to apply this formula is essential for solving complex trigonometric problems. It allows you to break down the problem step by step.

The cosine function, denoted as \( \cos(x) \), is a fundamental building block in trigonometry. It represents the horizontal coordinate of a point on the unit circle. The cosine function is periodic, meaning it repeats its values in regular intervals. Its period is \( 2\pi \). This can be especially useful when working with trigonometric identities and equations, like in your homework problem.
In your specific exercise, the term \( \cos^2(x) \) indicates the square of the cosine of angle \( x \). The simplified expression involves \( \frac{1}{2} \cos^2(x) \), making it easier to handle algebraically. Knowing these properties allows you to manipulate and solve equations more effectively.

Trigonometric equations are mathematical statements that equate trigonometric expressions to a value. Solving them can involve using fundamental identities, like the double angle formula, or properties of trigonometric functions. In the given problem, we had an equation: \ \[ \frac{1}{2} \cos^{2} x + C = \frac{1}{4} \cos (2 x) \ \]
By recognizing and applying the double angle formula, we transformed the right-hand side to match a simpler form. After simplification, it becomes \ \[ \frac{1}{2} \cos^2(x) - \frac{1}{4} \ \]
Now, with both sides involving \( \cos^2(x) \), we could isolate \( C \) and solve for its value, which turned out to be \( -\frac{1}{4} \). Breaking down the steps systematically helps in understanding and solving trigonometric equations effectively.