if `tan^(-1)(1/x)+cos^(-1)(2/sqrt5)=pi/4` then x equals

Options

A. `1/sqrt5`
B. `4/sqrt5`
C. 1
D. 3

Answer

To solve the equation \( \tan^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) = \frac{\pi}{4} \), we will follow these steps:

### Step 1: Isolate the Inverse Cosine
We can rewrite the equation as:
\[ \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{4} - \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \]

### Step 2: Use the Identity for Inverse Functions
Using the identity \( \tan^{-1}(a) + \cos^{-1}(b) = \frac{\pi}{2} \) when \( a = \tan(\theta) \) and \( b = \cos(\theta) \), we can express:
\[ \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) = \theta \implies \cos(\theta) = \frac{2}{\sqrt{5}} \]

### Step 3: Find the Sine of the Angle
Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ \sin^2(\theta) = 1 - \left(\frac{2}{\sqrt{5}}\right)^2 = 1 - \frac{4}{5} = \frac{1}{5} \]
Thus, \( \sin(\theta) = \frac{1}{\sqrt{5}} \).

### Step 4: Express Tangent in Terms of Sine and Cosine
Now, we can find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}} = \frac{1}{2} \]

### Step 5: Relate to the Original Equation
Since \( \tan^{-1}\left(\frac{1}{x}\right) = \theta \) and \( \tan(\theta) = \frac{1}{2} \), we have:
\[ \frac{1}{x} = \frac{1}{2} \implies x = 2 \]

### Step 6: Verify the Solution
To verify, substitute \( x = 2 \) back into the original equation:
\[ \tan^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \]
Calculating \( \tan^{-1}\left(\frac{1}{2}\right) \) gives us the angle whose tangent is \( \frac{1}{2} \), and we already found \( \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \) corresponds to the same angle. Therefore, the equation holds true.

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{2} \]