If `"tan" (pi "cos" theta) = "cot"(pi "sin" theta),` then the value(s) of `"cos" (theta-(pi)/(4))`, is (are)

Options

A. `(1)/(2)`
B. `(1)/(sqrt(2))`
C. `+-(1)/(2sqrt(2))`
D. none of these

Answer

To solve the equation \( \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \), we can follow these steps:

### Step 1: Rewrite the equation using cotangent
We know that \( \cot(x) = \frac{1}{\tan(x)} \). Therefore, we can rewrite the equation as:
\[ \tan(\pi \cos \theta) = \frac{1}{\tan(\pi \sin \theta)} \]
This implies:
\[ \tan(\pi \cos \theta) \tan(\pi \sin \theta) = 1 \]

### Step 2: Use the identity for tangent
Using the identity \( \tan(A) \tan(B) = 1 \) leads us to:
\[ \pi \cos \theta + \pi \sin \theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
This simplifies to:
\[ \cos \theta + \sin \theta = \frac{1}{2} + n \]

### Step 3: Rearranging the equation
Rearranging gives us:
\[ \cos \theta + \sin \theta = 2n + \frac{1}{2} \]

### Step 4: Use the cosine of angle subtraction formula
We want to find \( \cos\left(\theta - \frac{\pi}{4}\right) \). Using the cosine subtraction formula:
\[ \cos\left(\theta - \frac{\pi}{4}\right) = \cos \theta \cos \frac{\pi}{4} + \sin \theta \sin \frac{\pi}{4} \]
Since \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), we have:
\[ \cos\left(\theta - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} (\cos \theta + \sin \theta) \]

### Step 5: Substitute the value of \( \cos \theta + \sin \theta \)
Substituting the value from Step 3:
\[ \cos\left(\theta - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \left(2n + \frac{1}{2}\right) \]

### Step 6: Simplify the expression
This simplifies to:
\[ \cos\left(\theta - \frac{\pi}{4}\right) = \frac{2n + \frac{1}{2}}{\sqrt{2}} = \frac{4n + 1}{2\sqrt{2}} \]

### Step 7: Find specific values of \( n \)
To find specific values, we can substitute \( n = 0 \) and \( n = -1 \):
- For \( n = 0 \):
\[ \cos\left(\theta - \frac{\pi}{4}\right) = \frac{1}{2\sqrt{2}} \]
- For \( n = -1 \):
\[ \cos\left(\theta - \frac{\pi}{4}\right) = \frac{-1}{2\sqrt{2}} \]

### Final Answer
Thus, the values of \( \cos\left(\theta - \frac{\pi}{4}\right) \) are:
\[ \frac{1}{2\sqrt{2}} \quad \text{and} \quad \frac{-1}{2\sqrt{2}} \]

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