Question

If \(x \cdot \tan 45^\circ \cdot \sin 30^\circ = \cos 30^\circ \cdot \cot 60^\circ\), then x is equal to

• A. \(\sqrt{3}\)
• B. \(\frac{1}{\sqrt{3}}\)
• C. 1
• D. \(\frac{1}{2}\)

Hint:

Memorizing the trigonometric table for standard angles (\(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)) is essential for scoring quick marks in board exams!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a trigonometric equation with standard angles and we need to solve for the unknown variable \(x\).

Step 2: Key Formula or Approach:
Recall the values of trigonometric ratios for standard angles:
- \(\tan 45^\circ = 1\)
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\cot 60^\circ = \frac{1}{\sqrt{3}}\)

Step 3: Detailed Explanation:

• Write down the original equation:
\[ x \cdot \tan 45^\circ \cdot \sin 30^\circ = \cos 30^\circ \cdot \cot 60^\circ \]

• Substitute the standard values into the equation:
\[ x \cdot (1) \cdot \left(\frac{1}{2}\right) = \left(\frac{\sqrt{3}}{2}\right) \cdot \left(\frac{1}{\sqrt{3}}\right) \]

• Simplify both sides:
LHS:
\[ \frac{x}{2} \]
RHS:
\[ \frac{\sqrt{3}}{2\sqrt{3}} = \frac{1}{2} \]

• Equate LHS and RHS:
\[ \frac{x}{2} = \frac{1}{2} \]
Multiply both sides by 2:
\[ x = 1 \]


Step 4: Final Answer:
The value of \(x\) is 1.