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: Work out the fixed right-hand side.
$\cos 30^\circ \cdot \cot 60^\circ = \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{3}} = \frac{1}{2}$, and $\tan 45^\circ \cdot \sin 30^\circ = 1 \times \frac{1}{2} = \frac{1}{2}$, so the equation reduces to $\frac{x}{2} = \frac{1}{2}$.
Step 2: Test the given options directly rather than solving algebraically.
For $x=\sqrt{3}$, $\frac{x}{2} \approx 0.87$, too big. For $x=\frac{1}{\sqrt{3}}$, $\frac{x}{2} \approx 0.29$, too small. For $x=\frac{1}{2}$, $\frac{x}{2}=0.25$, too small.
Step 3: Confirm the match.
For $x=1$, $\frac{x}{2} = \frac{1}{2}$, exactly matching the right side.
\[ \boxed{x = 1} \]