Question

The value of \(\left( \frac{1}{3}\cot^2 30^\circ - \frac{1}{2}\sec^2 60^\circ \right)\) is :

• A. \(- 1\)
• B. \(- 2\)
• C. \(\frac{5}{8}\)
• D. \(\frac{7}{8}\)

Hint:

If you forget the value of \(\sec 60^\circ\) or \(\cot 30^\circ\), convert them to basic sine and cosine functions.
Using \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{1/2} = 2\) and \(\cot 30^\circ = \frac{\cos 30^\circ}{\sin 30^\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}\) ensures you avoid mistakes.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to compute the numerical value of a trigonometric expression involving standard angles.
The expression is \(\frac{1}{3}\cot^2 30^\circ - \frac{1}{2}\sec^2 60^\circ\).
To solve this, we need to recall the standard values of \(\cot 30^\circ\) and \(\sec 60^\circ\) and substitute them into the expression.

Step 2: Key Formula or Approach:
We use the standard values from the trigonometric table:
\[ \cot 30^\circ = \sqrt{3} \]
\[ \sec 60^\circ = 2 \]
We will square these values and perform basic algebraic operations.

Step 3: Detailed Explanation:
1. We begin by identifying the given trigonometric ratios:
- The value of \(\cot 30^\circ\) is \(\sqrt{3}\).
- The value of \(\sec 60^\circ\) is \(2\).
2. Next, we compute the squares of these trigonometric ratios:
- \(\cot^2 30^\circ = (\sqrt{3})^2 = 3\)
- \(\sec^2 60^\circ = (2)^2 = 4\)
3. Now, substitute these squared values back into the original expression:
\[ \frac{1}{3}\cot^2 30^\circ - \frac{1}{2}\sec^2 60^\circ = \frac{1}{3}(3) - \frac{1}{2}(4) \]
4. Simplify each term individually:
- The first term simplifies to:
\[ \frac{1}{3} \times 3 = 1 \]
- The second term simplifies to:
\[ \frac{1}{2} \times 4 = 2 \]
5. Subtract the simplified terms to get the final result:
\[ 1 - 2 = -1 \]
6. Therefore, the value of the given expression is \(-1\).

Step 4: Final Answer:
The correct option is (A).