If `2 tan^(-1)(costheta) = tan^(-1)(2 cosec theta)`, then show that `theta = (pi)/(4)`, when n is an y integer.

Explanation

To solve the equation \( 2 \tan^{-1}(\cos \theta) = \tan^{-1}(2 \csc \theta) \) and show that \( \theta = \frac{\pi}{4} + n\pi \) for any integer \( n \), we can follow these steps:

### Step 1: Use the double angle formula for arctangent
We start with the left-hand side of the equation:
\[ 2 \tan^{-1}(\cos \theta) = \tan^{-1}\left(\frac{2 \cos \theta}{1 - \cos^2 \theta}\right) \]
This uses the identity \( 2 \tan^{-1}(x) = \tan^{-1}\left(\frac{2x}{1-x^2}\right) \).

### Step 2: Rewrite the equation
Now we can rewrite the original equation as:
\[ \tan^{-1}\left(\frac{2 \cos \theta}{1 - \cos^2 \theta}\right) = \tan^{-1}(2 \csc \theta) \]

### Step 3: Eliminate the arctangent
Since the arctangent function is one-to-one, we can equate the arguments:
\[ \frac{2 \cos \theta}{1 - \cos^2 \theta} = 2 \csc \theta \]

### Step 4: Substitute \( \csc \theta \)
Recall that \( \csc \theta = \frac{1}{\sin \theta} \), so we can rewrite the right-hand side:
\[ \frac{2 \cos \theta}{1 - \cos^2 \theta} = \frac{2}{\sin \theta} \]

### Step 5: Cross-multiply
Cross-multiplying gives:
\[ 2 \cos \theta \sin \theta = 2(1 - \cos^2 \theta) \]

### Step 6: Simplify the equation
This simplifies to:
\[ \cos \theta \sin \theta = 1 - \cos^2 \theta \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can substitute \( 1 - \cos^2 \theta \) with \( \sin^2 \theta \):
\[ \cos \theta \sin \theta = \sin^2 \theta \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ \cos \theta \sin \theta - \sin^2 \theta = 0 \]
Factoring out \( \sin \theta \):
\[ \sin \theta (\cos \theta - \sin \theta) = 0 \]

### Step 8: Solve for \( \theta \)
This gives us two cases:
1. \( \sin \theta = 0 \) which implies \( \theta = n\pi \) for any integer \( n \).
2. \( \cos \theta - \sin \theta = 0 \) which implies \( \cos \theta = \sin \theta \), leading to \( \theta = \frac{\pi}{4} + n\pi \) for any integer \( n \).

### Conclusion
Thus, we have shown that:
\[ \theta = \frac{\pi}{4} + n\pi \]
for any integer \( n \).