Question

Let α and β be such that α+β=π. If \(\cos\alpha=\frac{1}{\sqrt2}\), then the value of cot (β-α) is

• A. ∞
• B. 1
• C. \(\frac{1}{2}\)
• D. \(\frac{1}{4}\)
• E. 0

Answer 1

Answer: (E)

Given:

• \( \alpha + \beta = \pi \)
• \( \cos \alpha = \frac{1}{\sqrt{2}} \)

Step 1: Find \( \alpha \) from \( \cos \alpha = \frac{1}{\sqrt{2}} \): \[ \alpha = \frac{\pi}{4} \quad \text{(since cosine is positive in the first quadrant)} \]

Step 2: Determine \( \beta \) using \( \alpha + \beta = \pi \): \[ \beta = \pi - \alpha = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

Step 3: Calculate \( \beta - \alpha \): \[ \beta - \alpha = \frac{3\pi}{4} - \frac{\pi}{4} = \frac{\pi}{2} \]

Step 4: Compute \( \cot(\beta - \alpha) \): \[ \cot\left(\frac{\pi}{2}\right) = 0 \]

The correct answer is (E) 0.

Answer 2

Given: - \( \alpha + \beta = \pi \) - \( \cos \alpha = \frac{1}{\sqrt{2}} \) 

We are asked to find the value of \( \cot (\beta - \alpha) \). 

Step 1: Use the identity for \( \cot(\beta - \alpha) \) The identity for \( \cot (\beta - \alpha) \) is: \[ \cot(\beta - \alpha) = \frac{\cot \beta \cot \alpha + 1}{\cot \beta - \cot \alpha} \] But to find this, we need to determine \( \cot \alpha \) and \( \cot \beta \). 

Step 2: Calculate \( \cot \alpha \) We are given that \( \cos \alpha = \frac{1}{\sqrt{2}} \). To find \( \cot \alpha \), we need to know \( \sin \alpha \). 

Using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), we get: \[ \sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left( \frac{1}{\sqrt{2}} \right)^2 = 1 - \frac{1}{2} = \frac{1}{2} \] 

Thus, \( \sin \alpha = \frac{1}{\sqrt{2}} \). Now, \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1 \). Step 3: Use the fact that \( \alpha + \beta = \pi \) 

Since \( \alpha + \beta = \pi \), we have \( \beta = \pi - \alpha \). 

Therefore, \( \cot \beta \) can be written using the identity for \( \cot (\pi - \theta) \): \[ \cot (\pi - \alpha) = -\cot \alpha \] So, \( \cot \beta = -\cot \alpha = -1 \). 

Step 4: Calculate \( \cot (\beta - \alpha) \) Now that we know \( \cot \alpha = 1 \) and \( \cot \beta = -1 \), we can substitute these values into the identity: \[ \cot (\beta - \alpha) = \frac{\cot \beta \cot \alpha + 1}{\cot \beta - \cot \alpha} = \frac{(-1)(1) + 1}{-1 - 1} = \frac{-1 + 1}{-2} = \frac{0}{-2} = 0 \]