Problem 101 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that \(f(x)=\frac{2 x(\sin x+\tan x)}{2\left[\frac{x+2 \pi}{\pi}\right]-3}\) is an odd function.

Step-by-Step Solution

Answer

To prove that the given function is odd, we need to show that \(f(-x) = -f(x)\) for all values of \(x\). After substituting \(-x\) in place of \(x\) and simplifying the expression using trigonometric identities, we obtained \(f(-x) = -\frac{2x(\sin(x) + \tan(x))}{2\left[\frac{x+2\pi}{\pi}\right] - 3}\). Comparing this to \(-f(x)\), we found that \(f(-x) = -f(x)\), confirming that the given function is indeed an odd function.

Step 1: Finding f(-x)

We will start by substituting \(-x\) in place of \(x\) in the given function: \(f(-x) = \frac{2(-x)(\sin(-x) + \tan(-x))}{2\left[\frac{-x + 2\pi}{\pi}\right] - 3}\) Now, let's simplify this expression with the help of trigonometric identities.

Step 2: Simplifying Trigonometric Identities

Recall the following trigonometric identities: 1. \(\sin(-x) = -\sin(x)\) 2. \(\tan(-x) = -\tan(x)\) Now, we substitute these identities in the expression for \(f(-x)\): \(f(-x) = \frac{2(-x)(-\sin(x) - \tan(x))}{2\left[\frac{-x + 2\pi}{\pi}\right] - 3}\)

Step 3: Simplifying the Expression

Let's simplify the expression for \(f(-x)\) further: \(f(-x) = -\frac{2x(\sin(x) + \tan(x))}{-2\left[\frac{x-2\pi}{\pi}\right] + 3}\) Now, let's simplify the denominator of the expression: \(-2\left[\frac{x-2\pi}{\pi}\right] + 3 = 2\left[\frac{-(x-2\pi)}{\pi}\right] - 3 = 2\left[\frac{x+2\pi}{\pi}\right] - 3\) So, we have: \(f(-x) = -\frac{2x(\sin(x) + \tan(x))}{2\left[\frac{x+2\pi}{\pi}\right] - 3}\)

Step 4: Comparing f(-x) and -f(x)

Now, let's compare \(f(-x)\) to \(-f(x)\): \(-f(x) = -\left(\frac{2x(\sin(x) + \tan(x))}{2\left[\frac{x+2\pi}{\pi}\right] - 3}\right) = -\frac{2x(\sin(x) + \tan(x))}{2\left[\frac{x+2\pi}{\pi}\right] - 3}\) As we can see, \(f(-x) = -\frac{2x(\sin(x) + \tan(x))}{2\left[\frac{x+2\pi}{\pi}\right] - 3} = -f(x)\) Since \(f(-x) = -f(x)\) for all values of \(x\), the given function \(f(x) = \frac{2 x(\sin x+\tan x)}{2\left[\frac{x+2\pi}{\pi}\right]-3}\) is an odd function.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in proving the properties of mathematical functions. These identities create connections between different trigonometric functions, allowing us to simplify expressions and solve equations more easily. In this proof for an odd function, two vital trigonometric identities were used:

• Identity 1: \( \sin(-x) = -\sin(x) \). This tells us that the sine of a negative angle equals negative sine of the angle. This property is reflective of the odd symmetry of the sine function.
• Identity 2: \( \tan(-x) = -\tan(x) \). Similarly, this demonstrates that the tangent of a negative angle is the negative tangent of the angle, underscoring the tangent function's odd symmetry as well.

Utilizing these identities, we replaced \( \sin(-x) \) with \( -\sin(x) \) and \( \tan(-x) \) with \( -\tan(x) \). This substitution was key in simplifying \( f(-x) \) and hence in showing the odd function property of \( f(x) \). Understanding these foundational identities can greatly aid in grasping more complex trigonometric proofs.

Function Properties

Function properties, such as even and odd functions, are important in understanding the symmetry and behavior of graphs. An odd function is characterized by the relationship \( f(-x) = -f(x) \) for every \( x \) in the domain of \( f \). This means the function is symmetrical about the origin. In more intuitive terms, flipping the graph over both the x-axis and y-axis would result in the same graph.

In the given problem, we needed to prove that \( f(x) = \frac{2 x(\sin x+\tan x)}{2\left[\frac{x+2\pi}{\pi}\right]-3} \) is an odd function. By calculating \( f(-x) \) and showing it equals \( -f(x) \), we utilized this property to confirm the function's odd nature. Recognizing these properties is crucial in analyzing and transforming functions and can also reveal much about their graphical representations.

Mathematical Proof

Mathematical proof is a logical process that establishes the truth of a statement using a series of deductive reasoning steps. It's a core method in mathematics to verify whether something is universally true. In the exercise at hand, we proved the function is odd, which involves demonstrating that \( f(-x) = -f(x) \) for every \( x \).

The proof began by replacing \( x \) with \( -x \) in the original function to find \( f(-x) \). With the previously discussed trigonometric identities, the expression was simplified. We then confirmed the equivalence of \( f(-x) \) to \(-f(x)\) by directly comparing the expressions. Each step follows logically from the last, ensuring that no assumptions go unverified. This careful, step-by-step approach is typical in mathematical proofs. By practicing this structured methodology, students gain a deeper understanding of how mathematical logic and various properties come together to form sound, reliable reasoning.