Number of solution of the equation `4sin^(2)x+tan^(2)x+cot^(2)x+cosec^(2)x=6` in `[0,pi]` is

Options

A. 1
B. 2
C. 0
D. 4

Answer

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

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

### Step 2: Use trigonometric identities
We know the following 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: Simplify the equation
To simplify, let \(y = \sin^2 x\). Then, \(\cos^2 x = 1 - y\). The equation becomes:
\[ 4y + \frac{y}{1-y} + \frac{1-y}{y} + \frac{1}{y} - 6 = 0 \]

### Step 4: Combine terms
Combining the fractions:
\[ 4y + \frac{y^2 + 1 - y}{y(1-y)} + \frac{1}{y} - 6 = 0 \]
This simplifies to:
\[ 4y + \frac{y^2 + 1 - y + 1 - 6y}{y(1-y)} = 0 \]

### Step 5: Set the equation to zero
Now we can multiply through by \(y(1-y)\) to eliminate the denominator (noting that \(y \neq 0\) and \(y \neq 1\)):
\[ 4y^2(1-y) + y^2 + 1 - y - 6y = 0 \]

### Step 6: Solve the resulting polynomial
This leads to a polynomial in \(y\). We can rearrange and combine like terms:
\[ (4 - 6)y^2 + (1 - 4)y + 1 = 0 \]
This simplifies to:
\[ -2y^2 - 3y + 1 = 0 \]

### Step 7: Use the quadratic formula
We can apply the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(-2)(1)}}{2(-2)} = \frac{3 \pm \sqrt{9 + 8}}{-4} = \frac{3 \pm \sqrt{17}}{-4} \]

### Step 8: Find the roots
Calculating the roots:
\[ y_1 = \frac{3 + \sqrt{17}}{-4}, \quad y_2 = \frac{3 - \sqrt{17}}{-4} \]

### Step 9: Determine valid solutions
We need to check if these values of \(y\) (which represent \(\sin^2 x\)) are within the interval \([0, 1]\).

### Step 10: Find the corresponding angles
For valid \(y\) values, we can find \(x\) using:
\[ \sin x = \sqrt{y} \quad \text{and} \quad \sin x = -\sqrt{y} \text{ (not valid in } [0, \pi]) \]

### Step 11: Count the solutions
Each valid \(y\) corresponds to two angles in \([0, \pi]\) if \(\sqrt{y}\) is valid. We find that there are two valid solutions in total.

### Conclusion
Thus, the number of solutions of the equation \( 4\sin^2 x + \tan^2 x + \cot^2 x + \csc^2 x = 6 \) in the interval \([0, \pi]\) is **2**.