The number of solutions of the equation `sin^(5)theta + 1/(sintheta) = 1/(costheta) +cos^(5)theta` where `theta int (0, pi/2)`, is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To find the number of solutions of the equation

\[ \sin^5 \theta + \frac{1}{\sin \theta} = \frac{1}{\cos \theta} + \cos^5 \theta \]

where \(\theta \in (0, \frac{\pi}{2})\), we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation for clarity:

\[ \sin^5 \theta + \frac{1}{\sin \theta} - \frac{1}{\cos \theta} - \cos^5 \theta = 0 \]

### Step 2: Analyze the terms
We know that both \(\sin \theta\) and \(\cos \theta\) are positive in the interval \((0, \frac{\pi}{2})\). Therefore, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step 3: Apply AM-GM Inequality
Using AM-GM on the left side:

\[ \sin^5 \theta + \frac{1}{\sin \theta} \geq 2 \sqrt{\sin^5 \theta \cdot \frac{1}{\sin \theta}} = 2 \sqrt{\sin^4 \theta} \]

This simplifies to:

\[ \sin^5 \theta + \frac{1}{\sin \theta} \geq 2 \sin^2 \theta \]

Similarly, for the right side:

\[ \cos^5 \theta + \frac{1}{\cos \theta} \geq 2 \sqrt{\cos^5 \theta \cdot \frac{1}{\cos \theta}} = 2 \sqrt{\cos^4 \theta} = 2 \cos^2 \theta \]

### Step 4: Set up the inequality
Now we have:

\[ \sin^5 \theta + \frac{1}{\sin \theta} \geq 2 \sin^2 \theta \]
\[ \cos^5 \theta + \frac{1}{\cos \theta} \geq 2 \cos^2 \theta \]

### Step 5: Equate the inequalities
Since we need to find the number of solutions, we set the two inequalities equal to each other:

\[ 2 \sin^2 \theta = 2 \cos^2 \theta \]

This simplifies to:

\[ \sin^2 \theta = \cos^2 \theta \]

### Step 6: Solve for \(\theta\)
Taking the square root gives:

\[ \tan^2 \theta = 1 \implies \tan \theta = 1 \]

This implies:

\[ \theta = \frac{\pi}{4} \]

### Step 7: Check the interval
Since \(\frac{\pi}{4}\) lies within the interval \((0, \frac{\pi}{2})\), we confirm that it is a valid solution.

### Conclusion
Thus, the number of solutions of the given equation in the interval \((0, \frac{\pi}{2})\) is:

\[ \boxed{1} \]