Use the power series expansion of \(\cos x\) to show that $$ \cos \frac{x}{2}=1-\frac{x^{2}}{8}+\frac{x^{4}}{384}-\frac{x^{6}}{46080}+\cdots $$

Series expansion in mathematics is a powerful tool that allows us to approximate complex functions using simpler polynomials. In essence, it involves expressing a function as the sum of an infinite sequence of terms, called a series. This is particularly useful when dealing with functions that are difficult to solve analytically.

Consider a function that's tricky to integrate or differentiate directly; series expansion can simplify our life! By approximating this function to a certain degree of accuracy with a polynomial, a series expansion like the Taylor series or Maclaurin series can be used to make otherwise intractable calculus problems manageable.

For the cosine function, as we've seen in the problem above, the series expansion consists of alternating positive and negative terms of even powers of the variable, starting with a constant term (which is 1 for cosine). The progression follows a pattern dictated by both the factorial of the even number (denominator) and the alternating sign, an example of an alternating series. This series can be used to calculate cos(x) for any value of x, to any desired degree of precision.

Trigonometric functions are the backbone of studying angles and oscillatory phenomena not only in mathematics but also in physics and engineering. Functions like sine, cosine, and tangent are defined initially for angles in a right-angled triangle, but their definitions are extended to all real numbers using the unit circle.

The cosine function, which represents the x-coordinate of a point on the unit circle as it moves counterclockwise from the positive x-axis, is an even and periodic function. Cosine and other trigonometric functions can be expanded into power series, which allow for their values to be calculated with arbitrary precision and provide a vital tool for analyzing the behavior of these functions without relying solely on geometric interpretations.

Calculus is an indispensable part of the engineering toolkit, providing methods for modeling and solving problems that involve rates of change and accumulation. Engineers use calculus for a multitude of applications, ranging from analyzing forces in structures to understanding electrical circuits.

One fundamental application involves using series expansions like the one demonstrated for the cosine function. Power series can represent a wide variety of functions near a point and play a vital role in numerical analysis, a field that develops methods for approximating and solving mathematical problems via computational algorithms.

For an engineer, mastering these mathematical tools is crucial for designing systems that behave predictably under various conditions. Understanding and manipulating series expansions ensure engineers can develop approximations that maintain system stability and efficiency without the need for exhaustive real-world testing.

Mathematical proofs are structured arguments that establish the truth of a given statement using logic and previously established theorems. They are the foundation of mathematical rigor and provide a systematic way to verify that a result holds under the specified conditions.

In the context of our problem, the power series proof of the cosine function represents a type of direct proof. It's constructed by expanding the function into its series form and manipulating the expansion directly to arrive at the desired result. This proof demonstrates how the cosine of half an angle, or any angle, can be expressed in terms of a series that converges to the exact value of the function as more terms are included. Proofs are not just about arriving at an answer; they also provide deep insight into why a mathematical statement is true and often reveal connections between different areas of mathematics.