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Understanding the double-angle identity is crucial when dealing with trigonometric functions and their properties. In trigonometry, the double-angle identities are specific formulas that express trigonometric functions of double angles in terms of single angles. For instance, one of the key identities is for the sine function, which states that \( \text{sin}(2x) = 2\text{sin}(x)\text{cos}(x) \). This relationship simplifies the expression \(g(x) = \text{sin}(2x)\) by representing it as twice the product of \(\text{sin}(x)\) and \(\text{cos}(x)\), both of which are familiar and more manageable functions.

These identities are immensely useful in various mathematical fields, including calculus and differential equations, as well as in physics. By using this identity, the complexity of problems involving periodic functions can be significantly decreased, as seen in the exercise where \( \text{sin}(2x) \) is expressed in terms of other trigonometric functions, aiding in determining linear dependence.

A linearly dependent subset is a fundamental concept in linear algebra that often confuses students. If a set of vectors (or functions, as is the case in the exercise) is such that one member of the set can be written as a linear combination of the others, then that set is linearly dependent. In simple terms, this means that there is a redundancy within the set—some information is overlapping. This redundancy is identified by finding nontrivial solutions to the equation \(a_1v_1 + a_2v_2 + \.\.\. + a_nv_n = 0\), where at least one of the coefficients \(a_i\) is non-zero.

Applying this to the provided exercise, you see that constants were found (\(a=0, b=1, c=-2\)), not all zero, that when applied to functions \(f(x), g(x), h(x)\), resulted in the zero function. This proves that these functions are not independent of each other—there's a linear dependence amongst them, meaning they form a linearly dependent subset in the vector space of real-valued functions.

The concept of a vector space might evoke images of arrows or points in space, but it is much broader than that. A vector space can also consist of functions, as is the case with \(\mathbb{F}(\mathbb{R})\), the space of all real-valued functions. Here, functions act as vectors, and we can add them together and multiply by scalars, much like we do with traditional vectors. What's essential about vector spaces is that they abide by certain axioms, such as distributivity, associativity, and the existence of an additive identity (the zero vector, which, for functions, is the zero function that assigns the value zero to all elements in its domain).

Considering our current exercise, we demonstrate that the functions \(f(x), g(x), h(x)\) belong to a vector space by checking their linear dependence. The zero function plays a pivotal role here; by showing that a nontrivial combination of these functions results in the zero function, we also indicate that functions can be added together and multiplied by real numbers, satisfying the axioms of a vector space.