Question

The number of solutions of the equation \( \tan 3x = \cot x \) in \( x \in [0, 2\pi] \) is:

• A. 4
• B. 6
• C. 2
• D. 8

Hint:

When solving trigonometric equations involving multiple angles, look for identities or manipulate the equation to simplify and find solutions in the given interval.

Answer 1

The Correct Option is D

Step 1: Rewrite the equation using trigonometric identities.
We are given the equation: \[ \tan 3x = \cot x \] Recall that: \[ \cot x = \frac{1}{\tan x} \] So, the equation becomes: \[ \tan 3x = \frac{1}{\tan x} \] Now multiply both sides by \( \tan x \) (provided \( \tan x \neq 0 \)): \[ \tan 3x \cdot \tan x = 1 \]
Step 2: Solve the equation.
We need to solve the equation: \[ \tan 3x \cdot \tan x = 1 \] Using the identity for the tangent of multiple angles, we can solve for the general solutions of \( x \) within the interval \( [0, 2\pi] \).
Step 3: Count the number of solutions.
The equation \( \tan 3x \cdot \tan x = 1 \) has 8 distinct solutions in the interval \( [0, 2\pi] \).
Step 4: Conclusion.
Thus, the number of solutions is 8. The correct answer is (4).