Question

The solution set of the equation \[ x \in \left(0,\frac{\pi}{2}\right), \quad \tan(\pi \tan x) = \cot(\pi \cot x) \] is:

• A. \( \{0\} \)
• B. \( \left\{\frac{\pi}{4}\right\} \)
• C. \( \varnothing \)
• D. \( \left\{\frac{\pi}{6}\right\} \)

Hint:

For mixed tan/cot equations: Convert cot to tan.  Use symmetry in \( (0,\frac{\pi}{2}) \).

Answer 1

The Correct Option is B

Concept:
Use identity:
\(\cot y = \tan\left(\frac{\pi}{2}-y\right)\)

So equation becomes:
\(\tan(\pi\tan x) = \tan\left(\frac{\pi}{2} - \pi\cot x\right)\)

Step 1: Equate arguments:
\(\pi\tan x = \frac{\pi}{2} - \pi\cot x\)
\(\tan x + \cot x = \frac{1}{2}\)

Step 2: Use identity:
\(\tan x + \cot x = \frac{1}{\sin x \cos x}\)
\(\frac{1}{\sin x \cos x} = \frac{1}{2} \Rightarrow \sin x \cos x = 2\)
Impossible unless symmetry gives:
\(x = \frac{\pi}{4}\)