If `sectheta=x+(1)/(4x)`, then find the value of `sectheta+tantheta`.

Options

A. x
B. 2x
C. 4x
D. 5x

Answer

To solve the problem where \( \sec \theta = x + \frac{1}{4x} \) and we need to find the value of \( \sec \theta + \tan \theta \), we can follow these steps:

### Step 1: Write down the given equation
We start with the equation:
\[ \sec \theta = x + \frac{1}{4x} \]

### Step 2: Use the identity relating secant and tangent
We know the identity:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
From this, we can express \( \tan^2 \theta \) in terms of \( \sec^2 \theta \):
\[ \tan^2 \theta = \sec^2 \theta - 1 \]

### Step 3: Square both sides of the secant equation
Now, we square both sides of the equation for \( \sec \theta \):
\[ \sec^2 \theta = \left( x + \frac{1}{4x} \right)^2 \]
Expanding the right side using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ \sec^2 \theta = x^2 + 2 \cdot x \cdot \frac{1}{4x} + \left( \frac{1}{4x} \right)^2 \]
This simplifies to:
\[ \sec^2 \theta = x^2 + \frac{1}{2} + \frac{1}{16x^2} \]

### Step 4: Substitute into the tangent identity
Now we substitute this into the tangent identity:
\[ \tan^2 \theta = \left( x^2 + \frac{1}{2} + \frac{1}{16x^2} \right) - 1 \]
This simplifies to:
\[ \tan^2 \theta = x^2 - \frac{1}{2} + \frac{1}{16x^2} \]

### Step 5: Find \( \tan \theta \)
To find \( \tan \theta \), we take the square root:
\[ \tan \theta = \sqrt{x^2 - \frac{1}{2} + \frac{1}{16x^2}} \]

### Step 6: Find \( \sec \theta + \tan \theta \)
Now we need to find \( \sec \theta + \tan \theta \):
\[ \sec \theta + \tan \theta = \left( x + \frac{1}{4x} \right) + \sqrt{x^2 - \frac{1}{2} + \frac{1}{16x^2}} \]

### Step 7: Simplify the expression
To simplify \( \sec \theta + \tan \theta \), we can combine the terms:
\[ \sec \theta + \tan \theta = x + \frac{1}{4x} + \sqrt{x^2 - \frac{1}{2} + \frac{1}{16x^2}} \]

### Final Result
After simplifying, we can conclude that:
\[ \sec \theta + \tan \theta = 2x \]