Problem Statement
Evaluate without using trigonometric tables:
$\frac{\cos 70^\circ}{\sin 20^\circ} + \frac{\cos 55^\circ \csc 35^\circ}{\tan 10^\circ \tan 20^\circ \tan 60^\circ \tan 70^\circ \tan 80^\circ}$
Verified Solution & Marking Scheme
Complementary Angles in Term 1
$\cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ \implies \frac{\cos 70^\circ}{\sin 20^\circ} = 1$.
Evaluate Numerator in Term 2
$\cos 55^\circ = \sin 35^\circ \implies \cos 55^\circ \csc 35^\circ = \sin 35^\circ \cdot \frac{1}{\sin 35^\circ} = 1$.
Evaluate Denominator in Term 2
$\tan 10^\circ \tan 80^\circ = \tan 10^\circ \cot 10^\circ = 1$. $\tan 20^\circ \tan 70^\circ = 1$. $\tan 60^\circ = \sqrt{3}$. Product $= 1 \times 1 \times \sqrt{3} = \sqrt{3}$.
Combine Terms
$1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}}$