If p = tan2 x + cot2 x, then which one of the following is correct?
p ≥ 2
Understanding the Problem: Analyzing p = tan2 x + cot2 x
The question asks for the correct relationship between the expression $p = \tan^2 x + \cot^2 x$ and the value 2. We are given four options involving inequalities.
First, let's understand the terms involved. $\tan x$ is defined as $\frac{\sin x}{\cos x}$ and $\cot x$ is defined as $\frac{\cos x}{\sin x}$. For both $\tan x$ and $\cot x$ to be defined simultaneously, $\sin x \ne 0$ and $\cos x \ne 0$. This means $x$ cannot be an integer multiple of $\frac{\pi}{2}$.
The expression is the sum of squares: $\tan^2 x + \cot^2 x$. Since the square of any real number is non-negative, $\tan^2 x \ge 0$ and $\cot^2 x \ge 0$ for all $x$ where $\tan x$ and $\cot x$ are defined.
Method 1: Using AM-GM Inequality for tan2 x and cot2 x
The Arithmetic Mean (AM) - Geometric Mean (GM) inequality states that for any two non-negative real numbers $a$ and $b$, the following inequality holds:
$\frac{a+b}{2} \ge \sqrt{ab}$
Equality holds if and only if $a=b$.
Let $a = \tan^2 x$ and $b = \cot^2 x$. Since $\tan x$ and $\cot x$ are real numbers (when defined), their squares $\tan^2 x$ and $\cot^2 x$ are non-negative. We can apply the AM-GM inequality:
$\frac{\tan^2 x + \cot^2 x}{2} \ge \sqrt{\tan^2 x \cdot \cot^2 x}$
We know that $\tan x \cdot \cot x = \tan x \cdot \frac{1}{\tan x} = 1$, provided $\tan x \ne 0$. Since $\tan x \ne 0$ for $\cot x$ to be defined (and vice versa), this holds.
Substituting this into the inequality:
$\frac{\tan^2 x + \cot^2 x}{2} \ge \sqrt{(\tan x \cdot \cot x)^2}$
$\frac{\tan^2 x + \cot^2 x}{2} \ge \sqrt{1^2}$
$\frac{\tan^2 x + \cot^2 x}{2} \ge \sqrt{1}$
$\frac{\tan^2 x + \cot^2 x}{2} \ge 1$
Now, multiply both sides by 2:
$\tan^2 x + \cot^2 x \ge 2$
Since $p = \tan^2 x + \cot^2 x$, we have $p \ge 2$.
Method 2: Using Algebraic Identity for tan2 x + cot2 x
Consider the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$. We can rearrange this to get $a^2 + b^2 = (a-b)^2 + 2ab$.
Let $a = \tan x$ and $b = \cot x$. Then the expression $p = \tan^2 x + \cot^2 x$ can be written as:
$p = (\tan x - \cot x)^2 + 2(\tan x)(\cot x)$
As before, $\tan x \cdot \cot x = 1$ for values of $x$ where both are defined.
$p = (\tan x - \cot x)^2 + 2(1)$
$p = (\tan x - \cot x)^2 + 2$
We know that the square of any real number is non-negative. Therefore, $(\tan x - \cot x)^2 \ge 0$ for all $x$ where $\tan x$ and $\cot x$ are defined.
Adding 2 to both sides of the inequality $(\tan x - \cot x)^2 \ge 0$, we get:
$(\tan x - \cot x)^2 + 2 \ge 0 + 2$
$(\tan x - \cot x)^2 + 2 \ge 2$
Since $p = (\tan x - \cot x)^2 + 2$, we conclude that $p \ge 2$.
When is p equal to 2?
From both methods, we found that $p \ge 2$. The minimum value of $p$ is 2. This occurs when the equality in the inequalities holds.
- In the AM-GM method, equality holds when $\tan^2 x = \cot^2 x$. This means $\tan^2 x = \frac{1}{\tan^2 x}$, which implies $\tan^4 x = 1$. Since $\tan^2 x$ must be non-negative, $\tan^2 x = 1$. This occurs when $\tan x = 1$ or $\tan x = -1$. For example, when $x = \frac{\pi}{4}$, $\tan x = 1$ and $\cot x = 1$, so $\tan^2 x + \cot^2 x = 1^2 + 1^2 = 1+1 = 2$. When $x = \frac{3\pi}{4}$, $\tan x = -1$ and $\cot x = -1$, so $\tan^2 x + \cot^2 x = (-1)^2 + (-1)^2 = 1+1 = 2$.
- In the algebraic method, equality holds when $(\tan x - \cot x)^2 = 0$. This means $\tan x - \cot x = 0$, or $\tan x = \cot x$. This leads to the same condition $\tan^2 x = 1$.
So, $p$ can be equal to 2.
Conclusion on the Relationship between p and 2
Based on the derivations using both AM-GM inequality and algebraic manipulation, we found that $p = \tan^2 x + \cot^2 x$ is always greater than or equal to 2 for all values of $x$ where $\tan x$ and $\cot x$ are defined.
Therefore, the correct relationship is $p \ge 2$.
Checking the Given Options
Let's look at the options provided:
- $p \le 2$: This is incorrect, as we found $p$ must be 2 or greater.
- $p \ge 2$: This matches our conclusion.
- $p < 2$: This is incorrect, as $p$ cannot be less than 2.
- $p > 2$: This is incorrect, as $p$ can be equal to 2.
The only option that correctly represents the relationship is $p \ge 2$.
Important Questions from Relations between AM, GM, HM
If the product of n positive numbers is unity, then their sum is?
If \(a_1, a_2, a_3,...,a_n\) are positive real numbers whose product is a fixed number C, then the minimum value of \(a_1+a_2+...+a_n\) is
In an acute angled ΔABC, the least value of sec A + sec B + sec C is:
Let x be the HM and y be the GM of two positive numbers m and n. If 5x = 4y, then which one of the following is correct?
Consider the following statements:
1. cos θ + sec θ can never be equal to 1.5.
2. tan θ + cot θ can never be less than 2.
Which of the above statements is/are correct?
