In a `Delta ABC` right angled at B, if tan `A = (1)/( sqrt(3))` , find the value of cos A cos C - sin A sin C

Options

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

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Identify the given information
We are given that in triangle ABC, angle B is a right angle (90 degrees), and tan A = 1 / √3.

### Step 2: Find angle A
We know that:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}} \]
This corresponds to the angle A being 30 degrees, since:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus, we have:
\[ A = 30^\circ \]

### Step 3: Find angle C
Using the property of triangles that the sum of angles in a triangle is 180 degrees, we can find angle C:
\[ A + B + C = 180^\circ \]
Substituting the known values:
\[ 30^\circ + 90^\circ + C = 180^\circ \]
This simplifies to:
\[ C = 180^\circ - 120^\circ = 60^\circ \]

### Step 4: Calculate cos A, cos C, sin A, and sin C
Now we will find the trigonometric ratios for angles A and C:
- \( \cos A = \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos C = \cos 60^\circ = \frac{1}{2} \)
- \( \sin A = \sin 30^\circ = \frac{1}{2} \)
- \( \sin C = \sin 60^\circ = \frac{\sqrt{3}}{2} \)

### Step 5: Substitute into the expression
We need to find the value of \( \cos A \cos C - \sin A \sin C \):
\[ \cos A \cos C - \sin A \sin C = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{2}\right) - \left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \]

### Step 6: Simplify the expression
Calculating the expression:
\[ = \frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{4} = 0 \]

### Final Answer
Thus, the value of \( \cos A \cos C - \sin A \sin C \) is:
\[ \boxed{0} \]