The number of solution of `sec^(2) theta + cosec^(2) theta+2 cosec^(2) theta=8, 0 le theta le pi//2` is

Options

A. 4
B. 3
C. 0
D. 2

Explanation

To solve the equation \( \sec^2 \theta + \csc^2 \theta + 2 \csc^2 \theta = 8 \) for \( 0 \leq \theta \leq \frac{\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
The given equation can be rewritten using the definitions of secant and cosecant:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta}, \quad \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
Thus, the equation becomes:
\[ \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} + 2 \cdot \frac{1}{\sin^2 \theta} = 8 \]
This simplifies to:
\[ \frac{1}{\cos^2 \theta} + \frac{3}{\sin^2 \theta} = 8 \]

### Step 2: Combine the fractions
To combine the fractions, we can find a common denominator:
\[ \frac{\sin^2 \theta + 3 \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} = 8 \]
Cross-multiplying gives:
\[ \sin^2 \theta + 3 \cos^2 \theta = 8 \sin^2 \theta \cos^2 \theta \]

### Step 3: Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \)
We can substitute \( \cos^2 \theta = 1 - \sin^2 \theta \):
\[ \sin^2 \theta + 3(1 - \sin^2 \theta) = 8 \sin^2 \theta (1 - \sin^2 \theta) \]
This simplifies to:
\[ \sin^2 \theta + 3 - 3 \sin^2 \theta = 8 \sin^2 \theta - 8 \sin^4 \theta \]
\[ -2 \sin^2 \theta + 3 = 8 \sin^2 \theta - 8 \sin^4 \theta \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 8 \sin^4 \theta - 10 \sin^2 \theta + 3 = 0 \]
Let \( x = \sin^2 \theta \). The equation becomes:
\[ 8x^2 - 10x + 3 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 8 \cdot 3}}{2 \cdot 8} \]
\[ x = \frac{10 \pm \sqrt{100 - 96}}{16} \]
\[ x = \frac{10 \pm 2}{16} \]
This gives us two solutions:
\[ x_1 = \frac{12}{16} = \frac{3}{4}, \quad x_2 = \frac{8}{16} = \frac{1}{2} \]

### Step 6: Find the values of \( \theta \)
Since \( x = \sin^2 \theta \):
1. For \( \sin^2 \theta = \frac{3}{4} \):
\[ \sin \theta = \frac{\sqrt{3}}{2} \implies \theta = \frac{\pi}{3} \]
2. For \( \sin^2 \theta = \frac{1}{2} \):
\[ \sin \theta = \frac{1}{\sqrt{2}} \implies \theta = \frac{\pi}{4} \]

### Conclusion: Count the solutions
Both angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) lie within the interval \( [0, \frac{\pi}{2}] \). Therefore, the number of solutions is:
\[ \boxed{2} \]