For a real angle x, consider the trigonometric expression tan^2 x + cot^2 x. What is the least possible value that this expression can take?

Introduction / Context:
This trigonometry question asks for the minimum possible value of tan^2 x + cot^2 x. It tests your ability to convert the problem into a simpler algebraic form and then apply basic inequality ideas, such as the AM GM inequality, rather than trying random angle values.

Given Data / Assumptions:
- x is a real angle for which both tan x and cot x are defined, so tan x is non zero.
- We need the least possible value of tan^2 x + cot^2 x.

Concept / Approach:
Let t = tan^2 x. Then cot^2 x = 1 / tan^2 x = 1 / t, provided t is positive. The expression becomes t + 1 / t for t > 0. The well known inequality t + 1 / t >= 2 for all positive real t gives the minimum value directly, which is attained at t = 1, corresponding to tan^2 x = 1.

Step-by-Step Solution:
Set t = tan^2 x. Then t > 0 because a square is always non negative and tan x is non zero. Then cot^2 x = 1 / tan^2 x = 1 / t. So tan^2 x + cot^2 x = t + 1 / t. For any positive real t, we have the inequality t + 1 / t >= 2, with equality only when t = 1. Thus the minimum possible value of t + 1 / t is 2. This minimum occurs when tan^2 x = 1, which means tan x = 1 or tan x = -1. So there are many angles where the minimum is attained.

Verification / Alternative check:
Take x = 45 degrees, where tan x = 1. Then tan^2 x = 1 and cot^2 x = 1. The sum is 1 + 1 = 2. If you try another angle, such as x = 30 degrees, tan^2 30 is 1 / 3 and cot^2 30 is 3, giving 1 / 3 + 3 = 10 / 3 which is larger than 2. This confirms that 2 is indeed the least value.

Why Other Options Are Wrong:
Values such as 0 or 1 are impossible because both tan^2 x and cot^2 x are non negative and cannot both be zero. A value of 3 or 4 is achievable but not minimal according to the inequality t + 1 / t >= 2.

Common Pitfalls:
Some learners mistakenly try to differentiate t + 1 / t or plug in arbitrary angles instead of using the simple inequality. Others misidentify the minimum as 0 or 1 by incorrectly assuming that one of the terms can be zero, which is not allowed because tan x cannot be zero if cot x is defined. Always check domain conditions carefully.

Final Answer:
The least possible value of tan^2 x + cot^2 x is 2.