Proving That tan x cot x Can Never Equal 3/2

In this article, we will delve into the fascinating world of trigonometric identities and prove that the expression tan x cot x can never be equal to 3/2. We will explore this through rigorous mathematical reasoning and algebraic manipulation.

Introduction to Trigonometric Functions

First, let's briefly review the definitions of tan x and cot x. The tangent function tan x is defined as the ratio of the sine and cosine functions, while the cotangent function cot x is the reciprocal of the tangent function.

Expressing tan x and cot x in Terms of y

To simplify our proof, we will express tan x as a variable y, which allows us to rewrite the expression tan x cot x more easily. Let us set:

y tan x

Then, by definition,

cot x 1/tan x 1/y

Substituting these into the expression tan x cot x,

tan x cot x y * 1/y 1

Equation Derivation and Quadratic Formula

Now, let's consider the claim that tan x cot x 3/2. We will start by setting up the equation:

y * 1/y - 3/2 0

Multiplying through by 2y (assuming y ne 0), we get:

2y^2 - 3y - 2 0

To solve this quadratic equation, we can use the quadratic formula, which states:

y (-b ± sqrt(b^2 - 4ac)) / 2a

Here, a 2, b -3, and c -2. Substituting these values into the quadratic formula:

y (3 ± sqrt((-3)^2 - 4 * 2 * (-2))) / 4

Calculating the discriminant:

b^2 - 4ac 9 16 25

Note that the discriminant is 25, which is positive. Therefore, the quadratic equation has two real roots, which are:

y (3 ± sqrt(25)) / 4 (3 ± 5) / 4

This gives us two solutions:

y 2 or y -1/2

However, since we are dealing with real values and the expression tan x cot x simplifies to 1, it means that y 1. Therefore, the quadratic equation does not hold true for y 1, and hence, the original claim that tan x cot x 3/2 is false for all real x.

Alternative Proofs

Let's explore another approach by directly substituting the definitions of tan x and cot x.

Recall that:

tan x sin x / cos x

cot x cos x / sin x

Thus,

tan x cot x (sin x / cos x) * (cos x / sin x) 1

Now, suppose tan x cot x 3/2. Then we have:

(sin x / cos x) * (cos x / sin x) 3/2

Which simplifies to:

1 3/2

This is a clear contradiction. Hence, it is impossible for tan x cot x to equal 3/2.

Conclusion

In conclusion, we have rigorously proven that tan x cot x can never be equal to 3/2. The use of algebraic manipulation and the quadratic formula highlighted the impossibility of the claim. Understanding such trigonometric properties is essential for advanced mathematics and related fields.