91Ó°ÊÓ

In trigonometry, the Pythagorean identity is a fundamental relation between the sine and cosine functions. It is expressed as \( \cos^2 x + \sin^2 x = 1 \). This equation shows that the square of the cosine of an angle plus the square of the sine of that angle equals one.
This identity can be extremely useful in solving trigonometric equations, such as the problem presented, \( \cos^2 x = \sin^2 x \). By recognizing the identity \( \sin^2 x = 1 - \cos^2 x \), we can substitute it into the equation, transforming it into \( \cos^2 x = 1 - \cos^2 x \).
This allows us to simplify and solve for the values of \( \cos^2 x \), a crucial step in finding the solutions to trigonometric equations.

Interval notation is a concise way of writing subsets of the real number line. It is often used to specify the set of solutions for a variable, particularly in trigonometry and calculus. The notation consists of two numbers, which represent the endpoints of the interval, enclosed in either brackets \([ ]\) or parentheses \(( )\).
Brackets indicate that the endpoint is included in the interval, while parentheses indicate that it is not. For instance, the interval \([0, 2\pi)\) means all numbers from 0 to \(2\pi\), including 0 but not \(2\pi\).
In our problem, this form of notation specifies the range of possible angle solutions that satisfy the equation. This is particularly important in trigonometry because angles often repeat in a periodic manner. Specifying the interval helps ensure that the results are within a single cycle of the periodic function.

Finding angle solutions is a key aspect of solving trigonometric equations. Once the equation is simplified, the next step is to determine which angles satisfy this equation, given the specified interval.
In this exercise, after simplifying the equation and finding \( \cos x = \pm \frac{\sqrt{2}}{2} \), we identify the corresponding angles.

• For \( \cos x = \frac{\sqrt{2}}{2} \), the angles within \([0, 2\pi)\) are \( \frac{\pi}{4} \) and \( \frac{7\pi}{4} \).
• For \( \cos x = -\frac{\sqrt{2}}{2} \), the angles are \( \frac{3\pi}{4} \) and \( \frac{5\pi}{4} \).

Angle solutions take into account these specific points on the unit circle where the cosine values match the given value, while adhering to the constraints of the interval notation.