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Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They often simplify complex trigonometric expressions and help solve trigonometric equations.

One important identity is the double angle identity for sine, which states: \[ \sin 2x = 2 \sin x \cos x \].

This identity expresses the sine of a double angle in terms of the sine and cosine of the angle itself. In our problem, using this identity allows us to transform the equation \( \sin 2x - \sin x = 0 \) into \( 2\sin x \cos x - \sin x = 0 \).

This conversion simplifies the problem by expressing \( \sin 2x \) without the 2x term, thus making it easier to factor and solve the equation. Understanding and applying trigonometric identities is key to solving more intricate trigonometric equations in algebra and calculus.

Factoring is a mathematical process used to simplify expressions or solve equations. It involves finding common elements in all terms of an expression to reduce it to a simpler form.

In trigonometric equations, factoring is helpful in isolating terms, making it easier to find solutions. For instance, after applying the trigonometric identity, our equation turns into: \( 2 \sin x \cos x - \sin x = 0 \). Here, \( \sin x \) is a common factor.

By factoring it out, the equation becomes \( \sin x (2 \cos x - 1) = 0 \). This step is crucial because it breaks down the original equation into two simpler equations: \( \sin x = 0 \) and \( 2 \cos x - 1 = 0 \).

Each of these simpler equations can be solved separately to find the solutions for \( x \). Employing factoring not only simplifies expressions but also opens up straightforward paths to find solutions.

Solutions to trigonometric equations are often expressed in radians rather than degrees. Radians are a natural unit in mathematics for measuring angles, particularly in calculus and higher-level mathematics.

Radians provide a direct link between the radius of a circle and the arc length. Solving trigonometric equations in radians offers precise, real-valued solutions.

In our exercise, the equation \( \sin x = 0 \) translates to \( x = 0, \pi, 2\pi, \ldots \), covering multiples of \( \pi \). These represent the positions on the unit circle where the sine equals zero.

Similarly, for \( 2 \cos x - 1 = 0 \), solving gives \( \cos x = 0.5 \), which corresponds to angles \( x = \pi/3 \) and \( x = 5\pi/3 \) in the range \( 0 \leq x < 2\pi \).

Understanding how to translate solutions from the factored equation into positions on the unit circle in radians helps in mastering trigonometric problem-solving.