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: Set up the equation
Given:
\[ \tan^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) = \frac{\pi}{4} \]

### Step 2: Express \(\cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\) as an angle \(\theta\)
Let:
\[ \theta = \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \]
This implies:
\[ \cos \theta = \frac{2}{\sqrt{5}} \]

### Step 3: Construct a right triangle
From \(\cos \theta = \frac{2}{\sqrt{5}}\), we can construct a right triangle where:
- Adjacent side = 2
- Hypotenuse = \(\sqrt{5}\)

Using the Pythagorean theorem, we can find the opposite side:
\[ \text{Opposite side} = \sqrt{(\sqrt{5})^2 - 2^2} = \sqrt{5 - 4} = 1 \]

### Step 4: Find \(\tan \theta\)
Now, we can find \(\tan \theta\):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{2} \]

### Step 5: Rewrite the original equation
Substituting \(\tan \theta\) back into the equation:
\[ \tan^{-1}\left(\frac{1}{x}\right) + \tan^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{4} \]

### Step 6: Use the tangent addition formula
Using the formula for the tangent of the sum of angles:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \quad \text{if } ab < 1 \]
Here, \(a = \frac{1}{x}\) and \(b = \frac{1}{2}\). Therefore:
\[ \tan^{-1}\left(\frac{\frac{1}{x} + \frac{1}{2}}{1 - \frac{1}{x} \cdot \frac{1}{2}}\right) = \frac{\pi}{4} \]

### Step 7: Set the tangent equal to 1
Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we have:
\[ \frac{\frac{1}{x} + \frac{1}{2}}{1 - \frac{1}{2x}} = 1 \]

### Step 8: Cross-multiply and simplify
Cross-multiplying gives:
\[ \frac{1}{x} + \frac{1}{2} = 1 - \frac{1}{2x} \]
Multiplying through by \(2x\) to eliminate the fractions:
\[ 2 + x = 2x - 1 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 2 + 1 = 2x - x \]
\[ 3 = x \]

### Step 10: Conclusion
Thus, the value of \(x\) is:
\[ \boxed{3} \]