If `sintheta=1//sqrt""2` then `(tantheta+costheta)=`
A. `1//sqrt""2`
B. `2//sqrt""2`
C. `3//sqrt""2`
D. `(1+sqrt""2)//sqrt""2`

Options

A. B
B. D
C. A
D. C

Answer

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

1. **Given**: \(\sin \theta = \frac{1}{\sqrt{2}}\)

### Step 1: Determine the values of \(\cos \theta\) and \(\tan \theta\)

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[ \sin^2 \theta = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

Substituting into the identity:

\[ \frac{1}{2} + \cos^2 \theta = 1 \]

\[ \cos^2 \theta = 1 - \frac{1}{2} = \frac{1}{2} \]

Taking the square root:

\[ \cos \theta = \frac{1}{\sqrt{2}} \quad (\text{since } \theta \text{ is in the first quadrant}) \]

### Step 2: Calculate \(\tan \theta\)

\(\tan \theta\) is defined as:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

Substituting the values we found:

\[ \tan \theta = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1 \]

### Step 3: Calculate \( \tan \theta + \cos \theta \)

Now we can compute \( \tan \theta + \cos \theta \):

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

### Step 4: Find a common denominator

To add these fractions, we convert \(1\) to a fraction with a denominator of \(\sqrt{2}\):

\[ 1 = \frac{\sqrt{2}}{\sqrt{2}} \]

Thus:

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

### Step 5: Final Answer

The expression simplifies to:

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

Thus, the correct answer is:

**D. \(\frac{1 + \sqrt{2}}{\sqrt{2}}\)**

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