The least value of |a| for which `tan theta` and `cot theta` are the roots of the equation `x^(2)+ax+b=0`

Options

A. 2
B. 1
C. `1//2`
D. 0

Answer

To find the least value of |a| for which `tan θ` and `cot θ` are the roots of the equation `x^2 + ax + b = 0`, we can follow these steps:

### Step 1: Identify the roots
Let the roots of the equation be `tan θ` and `cot θ`.

### Step 2: Use the relationships of roots
From Vieta's formulas, we know:
- The sum of the roots (tan θ + cot θ) = -a
- The product of the roots (tan θ * cot θ) = b

Since `tan θ * cot θ = 1`, we have:
\[ b = 1 \]

### Step 3: Calculate the sum of the roots
Now, we calculate the sum of the roots:
\[ \tan θ + \cot θ = \frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ} = \frac{\sin^2 θ + \cos^2 θ}{\sin θ \cos θ} \]

Using the identity \( \sin^2 θ + \cos^2 θ = 1 \):
\[ \tan θ + \cot θ = \frac{1}{\sin θ \cos θ} \]

### Step 4: Relate the sum to 'a'
From Vieta's, we have:
\[ -a = \frac{1}{\sin θ \cos θ} \]
Thus,
\[ a = -\frac{1}{\sin θ \cos θ} \]

### Step 5: Use the double angle identity
We can express \( \sin θ \cos θ \) in terms of \( \sin 2θ \):
\[ \sin θ \cos θ = \frac{1}{2} \sin 2θ \]
So,
\[ a = -\frac{2}{\sin 2θ} \]

### Step 6: Determine the range of 'a'
The sine function has a maximum value of 1. Therefore, the minimum value of |a| occurs when \( \sin 2θ \) is maximized:
\[ |a| = \left| -\frac{2}{\sin 2θ} \right| \]
This gives:
\[ |a| = \frac{2}{|\sin 2θ|} \]

### Step 7: Find the minimum value of |a|
Since \( |\sin 2θ| \) can take values from 0 to 1, the maximum value of \( |\sin 2θ| \) is 1. Thus, the minimum value of |a| is:
\[ |a| = 2 \]

### Conclusion
The least value of |a| for which `tan θ` and `cot θ` are the roots of the equation `x^2 + ax + b = 0` is:
\[ \boxed{2} \]

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