Question

\(\frac{(\sin 30^\circ \cdot \sec 60^\circ + \cos 30^\circ \cdot \csc 60^\circ)}{(\sec 45^\circ \cdot \cot 45^\circ \cdot \csc 45^\circ)}=\)

• A. 2
• B. 1
• C. \(\sqrt{3}\)
• D. \(\sqrt{6}\)

Answer 1

The Correct Option is B

We use trigonometric identities:
\(\sin 30^\circ = \frac{1}{2}, \sec 60^\circ = 2, \cos 30^\circ = \frac{\sqrt{3}}{2}, \csc 60^\circ = \frac{2}{\sqrt{3}},\)
\(\sec 45^\circ = \sqrt{2}, \cot 45^\circ = 1, \csc 45^\circ = \sqrt{2}\)
Numerator: \[ \sin 30^\circ \cdot \sec 60^\circ + \cos 30^\circ \cdot \csc 60^\circ = \left(\frac{1}{2} \cdot 2\right) + \left(\frac{\sqrt{3}}{2} \cdot \frac{2}{\sqrt{3}}\right) = 1 + 1 = 2 \] Denominator: \[ \sec 45^\circ \cdot \cot 45^\circ \cdot \csc 45^\circ = \sqrt{2} \cdot 1 \cdot \sqrt{2} = 2 \] So, the expression becomes: \[ \frac{2}{2} = 1 \]

The correct option is (B): \(1\)