If `tan theta=(1)/sqrt(5)` and `theta` lies in the first quadrant, the value of `cos theta` is :

Options

A. `sqrt((5)/(2))`
B. `-sqrt((5)/(6))`
C. `(1)/sqrt(6)`
D. `-(1)/sqrt(6)`

Explanation

To find the value of \( \cos \theta \) given that \( \tan \theta = \frac{1}{\sqrt{5}} \) and \( \theta \) lies in the first quadrant, we can follow these steps:

### Step 1: Use the identity relating tangent and secant
We know that:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
Substituting the value of \( \tan \theta \):
\[ 1 + \left(\frac{1}{\sqrt{5}}\right)^2 = \sec^2 \theta \]

### Step 2: Calculate \( \tan^2 \theta \)
Calculating \( \tan^2 \theta \):
\[ \tan^2 \theta = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5} \]

### Step 3: Substitute and simplify
Now substitute \( \tan^2 \theta \) into the identity:
\[ 1 + \frac{1}{5} = \sec^2 \theta \]
This simplifies to:
\[ \frac{5}{5} + \frac{1}{5} = \sec^2 \theta \]
\[ \frac{6}{5} = \sec^2 \theta \]

### Step 4: Find \( \sec \theta \)
Taking the square root of both sides:
\[ \sec \theta = \sqrt{\frac{6}{5}} = \frac{\sqrt{6}}{\sqrt{5}} \]

### Step 5: Find \( \cos \theta \)
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can find \( \cos \theta \):
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{\sqrt{5}}{\sqrt{6}} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{\sqrt{5}}{\sqrt{6}} \]

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