If `2 tan ^(-1)(cos x )=tan ^(-1)(2 cosec x),` then the value of x is

Options

A. `(3pi)/(4)`
B. `(pi)/(4)`
C. `(pi)/(4)`
D. None of these

Explanation

To solve the equation \( 2 \tan^{-1}(\cos x) = \tan^{-1}(2 \csc x) \), we will follow these steps:

### Step 1: Use the formula for \( 2 \tan^{-1}(a) \)

The formula for \( 2 \tan^{-1}(a) \) is given by:

\[ 2 \tan^{-1}(a) = \tan^{-1}\left(\frac{2a}{1 - a^2}\right) \]

In our case, let \( a = \cos x \). Thus, we can rewrite the left-hand side:

\[ 2 \tan^{-1}(\cos x) = \tan^{-1}\left(\frac{2 \cos x}{1 - \cos^2 x}\right) \]

### Step 2: Simplify \( 1 - \cos^2 x \)

We know that \( 1 - \cos^2 x = \sin^2 x \). Therefore, we can substitute this into our expression:

\[ 2 \tan^{-1}(\cos x) = \tan^{-1}\left(\frac{2 \cos x}{\sin^2 x}\right) \]

### Step 3: Set the two sides equal

Now we equate the two sides of the original equation:

\[ \tan^{-1}\left(\frac{2 \cos x}{\sin^2 x}\right) = \tan^{-1}(2 \csc x) \]

### Step 4: Remove the \( \tan^{-1} \)

Since the tangent inverse function is one-to-one, we can remove it from both sides (assuming the arguments are in the valid range):

\[ \frac{2 \cos x}{\sin^2 x} = 2 \csc x \]

### Step 5: Rewrite \( \csc x \)

Recall that \( \csc x = \frac{1}{\sin x} \). Substituting this in gives us:

\[ \frac{2 \cos x}{\sin^2 x} = \frac{2}{\sin x} \]

### Step 6: Cross-multiply

Cross-multiplying leads to:

\[ 2 \cos x \cdot \sin x = 2 \sin^2 x \]

### Step 7: Simplify the equation

Dividing both sides by 2 (assuming \( \sin x \neq 0 \)) gives us:

\[ \cos x \cdot \sin x = \sin^2 x \]

### Step 8: Rearranging the equation

Rearranging this gives:

\[ \cos x \cdot \sin x - \sin^2 x = 0 \]

Factoring out \( \sin x \):

\[ \sin x (\cos x - \sin x) = 0 \]

### Step 9: Solve for \( x \)

This gives us two cases:

1. \( \sin x = 0 \) which implies \( x = n\pi \) for \( n \in \mathbb{Z} \).
2. \( \cos x - \sin x = 0 \) which implies \( \cos x = \sin x \). This occurs at \( x = \frac{\pi}{4} + n\pi \) for \( n \in \mathbb{Z} \).

### Step 10: Find the specific solution

Since we are looking for values in the range of \( 0 \) to \( 2\pi \), the specific solution from the second case is:

\[ x = \frac{\pi}{4} \]

Thus, the value of \( x \) is:

\[ \boxed{\frac{\pi}{4}} \]