If `3 sec A - 2 cos B = sqrt3 and angle B = 30 ^(@) ,` then the value of `cos (A - B ) ` is

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the problem step by step, we will follow the instructions given in the transcript and provide a clear mathematical solution.

### Given:
1. \( 3 \sec A - 2 \cos B = \sqrt{3} \)
2. \( B = 30^\circ \)

### Step 1: Substitute the value of \( \cos B \)
We know that \( \cos 30^\circ = \frac{\sqrt{3}}{2} \). Substitute this into the equation:

\[ 3 \sec A - 2 \left(\frac{\sqrt{3}}{2}\right) = \sqrt{3} \]

### Step 2: Simplify the equation
The equation simplifies to:

\[ 3 \sec A - \sqrt{3} = \sqrt{3} \]

### Step 3: Isolate \( \sec A \)
Add \( \sqrt{3} \) to both sides:

\[ 3 \sec A = \sqrt{3} + \sqrt{3} \]

This simplifies to:

\[ 3 \sec A = 2\sqrt{3} \]

### Step 4: Solve for \( \sec A \)
Divide both sides by 3:

\[ \sec A = \frac{2\sqrt{3}}{3} \]

### Step 5: Find \( \cos A \)
Since \( \sec A = \frac{1}{\cos A} \), we have:

\[ \cos A = \frac{3}{2\sqrt{3}} \]

To rationalize the denominator:

\[ \cos A = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \]

### Step 6: Determine \( A \)
The angle \( A \) for which \( \cos A = \frac{\sqrt{3}}{2} \) is:

\[ A = 30^\circ \]

### Step 7: Calculate \( \cos(A - B) \)
Now we need to find \( \cos(A - B) \):

\[ A - B = 30^\circ - 30^\circ = 0^\circ \]

Thus,

\[ \cos(0^\circ) = 1 \]

### Final Answer:
The value of \( \cos(A - B) \) is \( 1 \).

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