If `sec theta + tan theta = 2 + sqrt 5` then the value of `sin theta + cos theta ` is :
यदि `sec theta + tan theta =2+sqrt5 " हो, तो " sin theta+ cos theta` का मान क्या होगा?

Options

A. A)`sqrt5`
B. B)`7/sqrt5`
C. C)`1/sqrt5`
D. D)`3/sqrt5`

Explanation

To solve the problem, we start with the equation given:

\[ \sec \theta + \tan \theta = 2 + \sqrt{5} \]

### Step 1: Use the identity
We know from trigonometric identities that:

\[ \sec^2 \theta - \tan^2 \theta = 1 \]

This can be factored as:

\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \]

### Step 2: Substitute the known value
Let \( x = \sec \theta + \tan \theta \). From the problem, we have:

\[ x = 2 + \sqrt{5} \]

Now, we can find \( \sec \theta - \tan \theta \):

\[ \sec \theta - \tan \theta = \frac{1}{\sec \theta + \tan \theta} = \frac{1}{2 + \sqrt{5}} \]

### Step 3: Rationalize the denominator
To simplify \( \frac{1}{2 + \sqrt{5}} \), we multiply the numerator and denominator by the conjugate:

\[ \sec \theta - \tan \theta = \frac{1 \cdot (2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})} = \frac{2 - \sqrt{5}}{4 - 5} = 2 - \sqrt{5} \]

### Step 4: Set up the equations
Now we have two equations:

1. \( \sec \theta + \tan \theta = 2 + \sqrt{5} \)
2. \( \sec \theta - \tan \theta = 2 - \sqrt{5} \)

### Step 5: Solve for \( \sec \theta \) and \( \tan \theta \)
Adding these two equations:

\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = (2 + \sqrt{5}) + (2 - \sqrt{5}) \]

This simplifies to:

\[ 2\sec \theta = 4 \implies \sec \theta = 2 \]

Now, substituting \( \sec \theta = 2 \) into one of the equations to find \( \tan \theta \):

\[ 2 + \tan \theta = 2 + \sqrt{5} \implies \tan \theta = \sqrt{5} \]

### Step 6: Find \( \sin \theta \) and \( \cos \theta \)
Using the definitions of secant and tangent:

\[ \sec \theta = \frac{1}{\cos \theta} \implies \cos \theta = \frac{1}{\sec \theta} = \frac{1}{2} \]

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \implies \sqrt{5} = \frac{\sin \theta}{\frac{1}{2}} \implies \sin \theta = \sqrt{5} \cdot \frac{1}{2} = \frac{\sqrt{5}}{2} \]

### Step 7: Calculate \( \sin \theta + \cos \theta \)
Now, we can find \( \sin \theta + \cos \theta \):

\[ \sin \theta + \cos \theta = \frac{\sqrt{5}}{2} + \frac{1}{2} = \frac{\sqrt{5} + 1}{2} \]

### Final Answer
Thus, the value of \( \sin \theta + \cos \theta \) is:

\[ \frac{\sqrt{5} + 1}{2} \]