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The double angle formulas are essential trigonometric identities that allow us to express trigonometric functions of doubled angles, like \(2x\), in terms of single angles, like \(x\). These formulas can significantly simplify calculations and transformations in trigonometry.
There are three primary double angle formulas you should know:

• \(\sin(2x) = 2\sin(x)\cos(x)\)
• \(\cos(2x) = \cos^2(x) - \sin^2(x)\)
• \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\)

Each of these formulas involves manipulating expressions of \(x\) as opposed to \(2x\). In solving the problem with \(f(x) = 1 - \cos 2x\), the use of \(\cos(2x)\) can leverage the identity \(\cos(2x) = 1 - 2\sin^2(x)\), helping us transform the function to be expressed in terms of \(\sin(x)\).
This simplification often aids in further analytical work, such as solving equations or integrating functions.

The cosine function is one of the foundational trigonometric functions, commonly denoted as \(\cos(x)\). This function represents the x-coordinate of a point on the unit circle as it relates to a given angle, \(x\), in standard position. Cosine has several useful properties:

• The function is periodic with a period of \(2\pi\).
• Cosine is an even function: \(\cos(-x) = \cos(x)\).
• Values range between -1 and 1, inclusive.

In relation to the given exercise \(f(x) = 1 - \cos 2x\), the cosine function's double angle formula allows us to express \(\cos 2x\) in a form that can be further simplified using sine, emphasizing the interconnected nature of trigonometric functions.
Understanding the graph and attributes of the cosine function can be crucial in visualizing and solving trigonometric equations.

The sine function, denoted as \(\sin(x)\), is another critical trigonometric function that represents the y-coordinate of a point on the unit circle for a given angle, \(x\). Here are some key characteristics:

• It is periodic with a period of \(2\pi\).
• Sine is an odd function: \(\sin(-x) = -\sin(x)\).
• The values of sine range between -1 and 1, inclusive.

In the context of the exercise, \(f(x) = 2\sin^2 x\) emerged after applying the double angle formula and simplifying. This highlights the utility of switching between sine and cosine forms using identities.
The sine function's smooth wave-like graph helps in plotting and understanding broader applications, including sound waves, light waves, and harmonic motion.