If `xtan60^(@)+cos45^(@)=sec45^(@)` then the value of `(x^(2)+1)` is

Options

A. `(6)/(7)`
B. `(7)/(6)`
C. `(5)/(6)`
D. `(6)/(5)`

Answer

To solve the equation \( x \tan 60^\circ + \cos 45^\circ = \sec 45^\circ \), we will follow these steps:

### Step 1: Substitute the values of trigonometric functions
We know:
- \( \tan 60^\circ = \sqrt{3} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sec 45^\circ = \sqrt{2} \)

Substituting these values into the equation gives us:
\[ x \cdot \sqrt{3} + \frac{1}{\sqrt{2}} = \sqrt{2} \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( x \):
\[ x \cdot \sqrt{3} = \sqrt{2} - \frac{1}{\sqrt{2}} \]

### Step 3: Simplify the right-hand side
To simplify \( \sqrt{2} - \frac{1}{\sqrt{2}} \):
\[ \sqrt{2} - \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{2 - 1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 4: Substitute back into the equation
Now substituting back, we have:
\[ x \cdot \sqrt{3} = \frac{1}{\sqrt{2}} \]

### Step 5: Solve for \( x \)
To find \( x \):
\[ x = \frac{1}{\sqrt{2} \cdot \sqrt{3}} = \frac{1}{\sqrt{6}} \]

### Step 6: Find \( x^2 \)
Now, we calculate \( x^2 \):
\[ x^2 = \left(\frac{1}{\sqrt{6}}\right)^2 = \frac{1}{6} \]

### Step 7: Calculate \( x^2 + 1 \)
Finally, we need to find \( x^2 + 1 \):
\[ x^2 + 1 = \frac{1}{6} + 1 = \frac{1}{6} + \frac{6}{6} = \frac{7}{6} \]

### Conclusion
Thus, the value of \( x^2 + 1 \) is \( \frac{7}{6} \).

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