If ` tan(pi cos theta )= cot (pi sin theta ) ` ,then ` cos^(2)(theta -pi//4) ` is equal to

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \]
Recall that \( \cot(x) = \frac{1}{\tan(x)} \), so 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 \( \tan(A) \tan(B) = 1 \) implies \( A + B = \frac{\pi}{2} + n\pi \) for some integer \( n \). Therefore, we can write:
\[ \pi \cos \theta + \pi \sin \theta = \frac{\pi}{2} + n\pi \]
Dividing through by \( \pi \):
\[ \cos \theta + \sin \theta = \frac{1}{2} + n \]

### Step 3: Simplify the equation
For simplicity, we can consider the case when \( n = 0 \):
\[ \cos \theta + \sin \theta = \frac{1}{2} \]

### Step 4: Square both sides
To eliminate the sine and cosine, we can square both sides:
\[ (\cos \theta + \sin \theta)^2 = \left(\frac{1}{2}\right)^2 \]
Expanding the left side:
\[ \cos^2 \theta + 2 \cos \theta \sin \theta + \sin^2 \theta = \frac{1}{4} \]
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ 1 + 2 \cos \theta \sin \theta = \frac{1}{4} \]
Rearranging gives:
\[ 2 \cos \theta \sin \theta = \frac{1}{4} - 1 = -\frac{3}{4} \]
Thus:
\[ \cos \theta \sin \theta = -\frac{3}{8} \]

### Step 5: Find \( \cos^2(\theta - \frac{\pi}{4}) \)
Using the angle subtraction formula:
\[ \cos(\theta - \frac{\pi}{4}) = \cos \theta \cos \frac{\pi}{4} + \sin \theta \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}(\cos \theta + \sin \theta) \]
Substituting \( \cos \theta + \sin \theta = \frac{1}{2} \):
\[ \cos(\theta - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{4} \]
Now squaring this:
\[ \cos^2(\theta - \frac{\pi}{4}) = \left(\frac{\sqrt{2}}{4}\right)^2 = \frac{2}{16} = \frac{1}{8} \]

### Final Answer
Thus, we find:
\[ \cos^2(\theta - \frac{\pi}{4}) = \frac{1}{8} \]

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