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Trigonometric identities are mathematical equations that express one trigonometric function in terms of another. These identities are fundamental tools often used to simplify complex trigonometric expressions and solve equations. One of the key identities used in trigonometry is the double angle formula. The double angle formula, particularly for cosine, states that:

• \(\cos(2x) = 2\cos^2(x) - 1\)

This formula allows us to express the cosine of a double angle in terms of the cosine of a single angle. Understanding and applying trigonometric identities like this one can help to simplify equations where multiple angles are present. In the problem at hand, the identity was used to transform the equation \(\cos(2u) = \cos(2v)\) into an expression involving \(\cos^2(u)\) and \(\cos^2(v)\), making it easier to solve.

The cosine function is one of the fundamental trigonometric functions, important in mathematics for describing relations in geometry and oscillatory motion. It is often abbreviated as 'cos' and varies between -1 and 1 for all possible angles. The function relates an angle in a right triangle to the ratio of the adjacent side length over the hypotenuse. In circular terms, it corresponds to the x-coordinate of the point on the unit circle.

• Range: [-1, 1]
• Periodicity: 2\(\pi\) radians or 360 degrees

In our exercise, the cosine function was not only used directly but also indirectly through the double angle identity, showing how it interrelates with itself at different angles. By understanding these interrelations, such as through the transformation \(\cos(2x) = 2\cos^2(x) - 1\), students can better navigate through trigonometric equations and identities.

Equation simplification is a crucial step in solving mathematical problems, especially in trigonometry, where expressions often get complicated. Simplification involves reducing expressions to a form that is easier to work with or solve. This is often done by using identities, factoring, and performing algebraic manipulations.In this particular example, we begin with the equation \(2\cos^2(u) - 1 = 2\cos^2(v) - 1\) upon applying the double angle formula. The critical steps taken involved:

• Adding 1 to both sides to eliminate the subtraction
• Dividing by 2 to simplify coefficients
• Taking the square root to solve for the absolute cosine values

By following these steps, the equation reduced to \(|\cos u| = |\cos v|\), demonstrating equality of the absolute cosine values. Mastering these skills allows students to handle trigonometric equations more efficiently, revealing solutions that are not immediately obvious.