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 \( \tan x = \cot\left(\frac{\pi}{2} - x\right) \), we can rewrite the right-hand side:
\[ \cot(5\pi \sin \theta) = \cot\left(\frac{\pi}{2} - 5\pi \cos \theta\right) \]

### Step 2: Set the angles equal
Since the cotangent function is equal when the angles are equal (plus an integer multiple of \( \pi \)), we can set the arguments equal:
\[ 5\pi \sin \theta = \frac{\pi}{2} - 5\pi \cos \theta + n\pi \]
where \( n \) is any integer.

### Step 3: Rearrange the equation
Rearranging gives:
\[ 5\pi \sin \theta + 5\pi \cos \theta = n\pi + \frac{\pi}{2} \]
Factoring out \( 5\pi \) from the left side:
\[ 5\pi (\sin \theta + \cos \theta) = n\pi + \frac{\pi}{2} \]
Dividing through by \( \pi \):
\[ 5(\sin \theta + \cos \theta) = n + \frac{1}{2} \]

### Step 4: Isolate \( \sin \theta + \cos \theta \)
Rearranging gives:
\[ \sin \theta + \cos \theta = \frac{n + \frac{1}{2}}{5} \]

### Step 5: Determine the range of \( \sin \theta + \cos \theta \)
The maximum and minimum values of \( \sin \theta + \cos \theta \) can be found using the identity:
\[ \sin \theta + \cos \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
Thus, the range is:
\[ -\sqrt{2} \leq \sin \theta + \cos \theta \leq \sqrt{2} \]

### Step 6: Set up inequalities
We can set up inequalities based on the range:
\[ -\sqrt{2} \leq \frac{n + \frac{1}{2}}{5} \leq \sqrt{2} \]
Multiplying through by 5 gives:
\[ -5\sqrt{2} \leq n + \frac{1}{2} \leq 5\sqrt{2} \]
Subtracting \( \frac{1}{2} \):
\[ -5\sqrt{2} - \frac{1}{2} \leq n \leq 5\sqrt{2} - \frac{1}{2} \]

### Step 7: Calculate bounds for \( n \)
Calculating \( 5\sqrt{2} \):
\[ 5\sqrt{2} \approx 5 \times 1.414 = 7.07 \]
Thus:
\[ -5\sqrt{2} - \frac{1}{2} \approx -7.07 - 0.5 = -7.57 \]
and
\[ 5\sqrt{2} - \frac{1}{2} \approx 7.07 - 0.5 = 6.57 \]

### Step 8: Determine integer values for \( n \)
The integer values of \( n \) satisfying:
\[ -7.57 \leq n \leq 6.57 \]
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 number of solutions for the equation \( \cot(5\pi \sin \theta) = \tan(5\pi \cos \theta) \) in the interval \( (0, 2\pi) \) is:
\[ \boxed{14} \]