The set of values of `lambda in R` such that `tan^2 theta + sec theta = lambda` holds for some `theta` is

Answer

To find the set of values of \( \lambda \in \mathbb{R} \) such that \( \tan^2 \theta + \sec \theta = \lambda \) holds for some \( \theta \), we can follow these steps:

### Step 1: Use the identity for \( \tan^2 \theta \)
We know that:
\[ \tan^2 \theta + 1 = \sec^2 \theta \]
Thus, we can express \( \tan^2 \theta \) in terms of \( \sec^2 \theta \):
\[ \tan^2 \theta = \sec^2 \theta - 1 \]

### Step 2: Substitute into the equation
Substituting \( \tan^2 \theta \) into the original equation gives:
\[ \sec^2 \theta - 1 + \sec \theta = \lambda \]
This simplifies to:
\[ \sec^2 \theta + \sec \theta - 1 = \lambda \]

### Step 3: Rearranging the equation
Rearranging the equation, we have:
\[ \lambda = \sec^2 \theta + \sec \theta - 1 \]

### Step 4: Completing the square
To analyze the expression \( \sec^2 \theta + \sec \theta - 1 \), we can complete the square:
\[ \lambda = \left(\sec \theta + \frac{1}{2}\right)^2 - \frac{1}{4} - 1 \]
This simplifies to:
\[ \lambda = \left(\sec \theta + \frac{1}{2}\right)^2 - \frac{5}{4} \]

### Step 5: Finding the range of \( \lambda \)
The term \( \left(\sec \theta + \frac{1}{2}\right)^2 \) is always non-negative, and its minimum value occurs when \( \sec \theta = -\frac{1}{2} \) (which is not possible since \( \sec \theta \) cannot be between -1 and 1). Therefore, we consider the minimum value of \( \sec \theta \):

1. The range of \( \sec \theta \) is \( (-\infty, -1] \cup [1, \infty) \).
2. Thus, the minimum value of \( \sec \theta + \frac{1}{2} \) occurs as \( \sec \theta \to -1 \) or \( \sec \theta \to 1 \):
- When \( \sec \theta \to -1 \): \( \sec \theta + \frac{1}{2} \to -\frac{1}{2} \)
- When \( \sec \theta \to 1 \): \( \sec \theta + \frac{1}{2} \to \frac{3}{2} \)

### Step 6: Calculating the minimum value of \( \lambda \)
Thus, the minimum value of \( \left(\sec \theta + \frac{1}{2}\right)^2 \) is:
\[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]
Substituting this back into the equation for \( \lambda \):
\[ \lambda_{\text{min}} = \frac{1}{4} - \frac{5}{4} = -1 \]

### Step 7: Conclusion
Since \( \left(\sec \theta + \frac{1}{2}\right)^2 \) can take any non-negative value, \( \lambda \) can take values from \( -1 \) to \( \infty \):
\[ \lambda \geq -1 \]

Thus, the set of values of \( \lambda \) such that \( \tan^2 \theta + \sec \theta = \lambda \) holds for some \( \theta \) is:
\[ \lambda \in [-1, \infty) \]

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