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The double-angle formula is a trigonometric identity that expresses trigonometric functions of double angles in terms of single angles. This is useful for simplifying expressions involving trigonometric functions at angles like \(2x\). For the cosine function, the double-angle formula is \(\cos 2x = \cos^2 x - \sin^2 x\). By substituting \(\cos 2x\), we can also write it as \(\cos^2 2x = (\cos 2x)^2 = (1 - 2\sin^2 x)^2 = 1 - 2\sin^2 x\). This reformulation helps to simplify and solve trigonometric inequalities and equations.

In the context of the original exercise, starting with \(\cos^{2} 2x + \cos^{2} x \leq 1\), using the double-angle formula for \(\cos^{2} 2x\) allows us to rephrase the expression in terms of \(\sin x\). This not only makes the expression easier to handle but also facilitates the application of other identities like the Pythagorean identity.

The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that for any angle \(x\), \(\sin^2 x + \cos^2 x = 1\). This identity comes from the Pythagorean theorem applied to the unit circle, where the radius is 1.

In solving trigonometric inequalities like the one given in the exercise, the Pythagorean identity helps us substitute \(\cos^2 x\) with \(1 - \sin^2 x\) or vice versa. By doing this, we often simplify the expressions, making them more manageable, as evident when we transform \(1 - 2\sin^2 x + \cos^2 x \leq 1\) into \(-\sin^2 x \leq 0\).

The Pythagorean identity is key to reduce complex expressions, allowing us to focus directly on solving simplified inequalities.

Simplifying inequalities involves reducing a given inequality to a form that is easier to understand and solve. The process often includes using algebraic manipulation along with identities such as the double-angle formula and Pythagorean identity.

In the solution for the exercise \(\cos^{2} 2x + \cos^{2} x \leq 1\), simplifying plays a crucial role:

• First, rewrite \(\cos^{2} 2x\) using the double-angle formula.
• Then, apply the Pythagorean identity to express \(\cos^{2} x\) as \(1 - \sin^2 x\).
• These substitutions result in \(-\sin^{2} x \leq 0\), an inequality that is simpler and more intuitive.

Once simplified, the solution involves understanding that \(\sin^2 x\) is always nonnegative, meaning it is \(\geq 0\). This informs us that the inequality holds for all \(x\), showing how important simplification is in determining the solution set of an inequality.