The number of solutions of the equation `4sin^(2)x+tan^(2)x+cot^(2)x+"cosec"^(2)x=6" in " [0, 2pi]` :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \(4\sin^2 x + \tan^2 x + \cot^2 x + \csc^2 x = 6\) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation:
\[ 4\sin^2 x + \tan^2 x + \cot^2 x + \csc^2 x - 6 = 0 \]

### Step 2: Substitute identities
Recall the trigonometric identities:
- \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\)
- \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\)
- \(\csc^2 x = \frac{1}{\sin^2 x}\)

Substituting these identities into the equation gives:
\[ 4\sin^2 x + \frac{\sin^2 x}{\cos^2 x} + \frac{\cos^2 x}{\sin^2 x} + \frac{1}{\sin^2 x} - 6 = 0 \]

### Step 3: Combine terms
To simplify, we can multiply through by \(\sin^2 x \cos^2 x\) to eliminate the fractions. However, a more straightforward approach is to isolate the terms involving \(\tan\) and \(\cot\):
\[ 4\sin^2 x + \tan^2 x + \cot^2 x + \csc^2 x = 6 \]

### Step 4: Group the terms
We can group the terms as follows:
\[ (2\sin x - \cos x)^2 + (\tan x - \cot x)^2 = 0 \]

### Step 5: Set each square to zero
For the sum of squares to equal zero, each square must be zero:
1. \(2\sin x - \cos x = 0\)
2. \(\tan x - \cot x = 0\)

### Step 6: Solve the first equation
From \(2\sin x - \cos x = 0\):
\[ 2\sin x = \cos x \implies \tan x = \frac{1}{2} \]
The solutions for \(\tan x = \frac{1}{2}\) in the interval \([0, 2\pi]\) are:
\[ x = \tan^{-1}\left(\frac{1}{2}\right) \quad \text{and} \quad x = \tan^{-1}\left(\frac{1}{2}\right) + \pi \]

### Step 7: Solve the second equation
From \(\tan x - \cot x = 0\):
\[ \tan^2 x = 1 \implies \tan x = 1 \quad \text{or} \quad \tan x = -1 \]
The solutions for \(\tan x = 1\) are:
\[ x = \frac{\pi}{4} + n\pi \quad (n = 0, 1) \]
This gives:
\[ x = \frac{\pi}{4}, \frac{5\pi}{4} \]
The solutions for \(\tan x = -1\) are:
\[ x = \frac{3\pi}{4}, \frac{7\pi}{4} \]

### Step 8: Collect all solutions
Now we collect all the solutions:
1. From \(2\sin x - \cos x = 0\): \(x = \tan^{-1}\left(\frac{1}{2}\right), \tan^{-1}\left(\frac{1}{2}\right) + \pi\)
2. From \(\tan x - \cot x = 0\): \(x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\)

### Step 9: Count the unique solutions
We have:
- Two solutions from the first equation.
- Four solutions from the second equation.

### Conclusion
Thus, the total number of unique solutions in the interval \([0, 2\pi]\) is **4**.