If `tan^(- 1)\ 1/4+tan^(- 1)\ 2/9=1/2cos^(- 1)x` then `x` is equal to

Options

A. (a)`(1)/(2)`
B. (b)`(2)/(5)`
C. (c)`(3)/(5)`
D. (d)None of these

Answer

To solve the equation \( \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) = \frac{1}{2} \cos^{-1}(x) \), we will follow these steps:

### Step 1: Use the identity for the sum of inverse tangents
We know that:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]
if \( ab < 1 \).

Here, let \( a = \frac{1}{4} \) and \( b = \frac{2}{9} \).

### Step 2: Calculate \( a + b \) and \( ab \)
Calculate:
\[ a + b = \frac{1}{4} + \frac{2}{9} \]
To add these fractions, find a common denominator. The least common multiple of 4 and 9 is 36.
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{2}{9} = \frac{8}{36} \]
So,
\[ a + b = \frac{9}{36} + \frac{8}{36} = \frac{17}{36} \]

Now calculate \( ab \):
\[ ab = \frac{1}{4} \cdot \frac{2}{9} = \frac{2}{36} = \frac{1}{18} \]

### Step 3: Apply the identity
Now substitute into the identity:
\[ \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) = \tan^{-1}\left(\frac{\frac{17}{36}}{1 - \frac{1}{18}}\right) \]

Calculate \( 1 - ab \):
\[ 1 - ab = 1 - \frac{1}{18} = \frac{18 - 1}{18} = \frac{17}{18} \]

Now substitute back into the equation:
\[ \tan^{-1}\left(\frac{\frac{17}{36}}{\frac{17}{18}}\right) = \tan^{-1}\left(\frac{17 \cdot 18}{36 \cdot 17}\right) = \tan^{-1}\left(\frac{18}{36}\right) = \tan^{-1}\left(\frac{1}{2}\right) \]

### Step 4: Set the equation
Now we have:
\[ \tan^{-1}\left(\frac{1}{2}\right) = \frac{1}{2} \cos^{-1}(x) \]

### Step 5: Use the double angle identity for cosine
We can use the identity:
\[ 2 \tan^{-1}(y) = \cos^{-1}\left(\frac{1 - y^2}{1 + y^2}\right) \]
Here, \( y = \frac{1}{2} \):
\[ 2 \tan^{-1}\left(\frac{1}{2}\right) = \cos^{-1}\left(\frac{1 - \left(\frac{1}{2}\right)^2}{1 + \left(\frac{1}{2}\right)^2}\right) \]
Calculating:
\[ 1 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4} \]
\[ 1 + \left(\frac{1}{2}\right)^2 = 1 + \frac{1}{4} = \frac{5}{4} \]
Thus:
\[ \cos^{-1}\left(\frac{\frac{3}{4}}{\frac{5}{4}}\right) = \cos^{-1}\left(\frac{3}{5}\right) \]

### Step 6: Compare both sides
Now we have:
\[ \frac{1}{2} \cos^{-1}(x) = \frac{1}{2} \cos^{-1}\left(\frac{3}{5}\right) \]

### Step 7: Solve for \( x \)
By comparing both sides, we can conclude:
\[ x = \frac{3}{5} \]

### Final Answer
Thus, the value of \( x \) is \( \frac{3}{5} \).

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