Solve for x, if : `tan (cos^(-1)x)=(2)/(sqrt(5))`

Answer

To solve the equation \( \tan(\cos^{-1} x) = \frac{2}{\sqrt{5}} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan(\cos^{-1} x) = \frac{2}{\sqrt{5}} \]

### Step 2: Let \( \theta = \cos^{-1} x \)
From the equation, we can set:
\[ \theta = \cos^{-1} x \]
Thus, we have:
\[ \tan \theta = \frac{2}{\sqrt{5}} \]

### Step 3: Create a right triangle
Using the definition of tangent, we can represent this as a right triangle where:
- The opposite side (perpendicular) is 2
- The adjacent side (base) is \( \sqrt{5} \)

### Step 4: Find the hypotenuse using Pythagorean theorem
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the values:
\[ h^2 = 2^2 + (\sqrt{5})^2 = 4 + 5 = 9 \]
Thus, the hypotenuse \( h \) is:
\[ h = \sqrt{9} = 3 \]

### Step 5: Find \( \cos \theta \)
Now, we can find \( \cos \theta \):
\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{\sqrt{5}}{3} \]

### Step 6: Relate back to \( x \)
Since \( \theta = \cos^{-1} x \), we have:
\[ x = \cos(\theta) = \frac{\sqrt{5}}{3} \]

### Final Answer
Thus, the solution for \( x \) is:
\[ \boxed{\frac{\sqrt{5}}{3}} \]

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