The number of solutions of `cot(5pi sin theta )=tan (5 pi cos theta ), AA theta in (0,2pi)` is

Options

A. 7
B. 14
C. 21
D. 28

Answer

To solve the equation \( \cot(5\pi \sin \theta) = \tan(5\pi \cos \theta) \) for \( \theta \) in the interval \( (0, 2\pi) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cot(5\pi \sin \theta) = \tan(5\pi \cos \theta) \]
Using the identity \( \cot x = \frac{1}{\tan x} \), we can rewrite the equation as:
\[ \frac{1}{\tan(5\pi \sin \theta)} = \tan(5\pi \cos \theta) \]
This implies:
\[ \tan(5\pi \sin \theta) \tan(5\pi \cos \theta) = 1 \]

### Step 2: Use the tangent identity
From the identity \( \tan A \tan B = 1 \), we know that:
\[ 5\pi \sin \theta + 5\pi \cos \theta = \frac{\pi}{2} + n\pi \]
for some integer \( n \).

### Step 3: Simplify the equation
Dividing the entire equation by \( 5\pi \):
\[ \sin \theta + \cos \theta = \frac{1}{10} + \frac{n}{5} \]

### Step 4: Analyze the range of \( \sin \theta + \cos \theta \)
The maximum value of \( \sin \theta + \cos \theta \) occurs at \( \theta = \frac{\pi}{4} \) and is given by:
\[ \sqrt{2} \]
The minimum value occurs at \( \theta = \frac{5\pi}{4} \) and is:
\[ -\sqrt{2} \]
Thus, we have:
\[ -\sqrt{2} \leq \sin \theta + \cos \theta \leq \sqrt{2} \]

### Step 5: Set up inequalities
We need to find the integer values of \( n \) such that:
\[ -\sqrt{2} \leq \frac{1}{10} + \frac{n}{5} \leq \sqrt{2} \]

### Step 6: Solve the inequalities
1. For the left inequality:
\[ -\sqrt{2} \leq \frac{1}{10} + \frac{n}{5} \]
Multiplying through by \( 10 \):
\[ -10\sqrt{2} \leq 1 + 2n \]
This simplifies to:
\[ 2n \geq -10\sqrt{2} - 1 \quad \Rightarrow \quad n \geq \frac{-10\sqrt{2} - 1}{2} \]

2. For the right inequality:
\[ \frac{1}{10} + \frac{n}{5} \leq \sqrt{2} \]
Multiplying through by \( 10 \):
\[ 1 + 2n \leq 10\sqrt{2} \]
This simplifies to:
\[ 2n \leq 10\sqrt{2} - 1 \quad \Rightarrow \quad n \leq \frac{10\sqrt{2} - 1}{2} \]

### Step 7: Calculate bounds for \( n \)
Calculating the numerical values:
- \( \sqrt{2} \approx 1.414 \)
- Thus, \( 10\sqrt{2} \approx 14.14 \)

So we have:
1. Lower bound:
\[ n \geq \frac{-10(1.414) - 1}{2} \approx \frac{-15.14}{2} \approx -7.57 \]
2. Upper bound:
\[ n \leq \frac{14.14 - 1}{2} \approx \frac{13.14}{2} \approx 6.57 \]

### Step 8: Determine integer values of \( n \)
The integer values of \( n \) are:
\[ n = -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \]
This gives us a total of \( 14 \) integer values.

### Conclusion
Thus, the total number of solutions for the equation \( \cot(5\pi \sin \theta) = \tan(5\pi \cos \theta) \) in the interval \( (0, 2\pi) \) is:
\[ \boxed{14} \]