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
The given equation can be simplified as follows:
\[ \sec^2 \theta + 3 \csc^2 \theta = 8 \]

### Step 2: Substitute trigonometric identities
We know that:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \quad \text{and} \quad \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
Substituting these into the equation gives:
\[ \frac{1}{\cos^2 \theta} + 3 \cdot \frac{1}{\sin^2 \theta} = 8 \]

### Step 3: Find a common denominator
The common denominator for the left-hand side is \( \cos^2 \theta \sin^2 \theta \):
\[ \frac{\sin^2 \theta + 3 \cos^2 \theta}{\cos^2 \theta \sin^2 \theta} = 8 \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ \sin^2 \theta + 3 \cos^2 \theta = 8 \cos^2 \theta \sin^2 \theta \]

### Step 5: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ (1 - \cos^2 \theta) + 3 \cos^2 \theta = 8 \cos^2 \theta (1 - \cos^2 \theta) \]
This simplifies to:
\[ 1 + 2 \cos^2 \theta = 8 \cos^2 \theta - 8 \cos^4 \theta \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 8 \cos^4 \theta - 6 \cos^2 \theta + 1 = 0 \]

### Step 7: Let \( x = \cos^2 \theta \)
Let \( x = \cos^2 \theta \). The equation becomes:
\[ 8x^2 - 6x + 1 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 8 \cdot 1}}{2 \cdot 8} \]
\[ x = \frac{6 \pm \sqrt{36 - 32}}{16} \]
\[ x = \frac{6 \pm 2}{16} \]
This gives two solutions:
1. \( x = \frac{8}{16} = \frac{1}{2} \)
2. \( x = \frac{4}{16} = \frac{1}{4} \)

### Step 9: Find \( \theta \)
Now substituting back for \( \cos^2 \theta \):
1. \( \cos^2 \theta = \frac{1}{2} \) gives \( \theta = \frac{\pi}{4} \)
2. \( \cos^2 \theta = \frac{1}{4} \) gives \( \theta = \frac{\pi}{3} \)

### Step 10: Count the solutions
Both solutions \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{\pi}{3} \) lie within the interval \( [0, \frac{\pi}{2}] \).

### Final Answer
Thus, the number of solutions is **2**.

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