Question

The value of $ \frac {(cos 20^\circ + sin 20^\circ )(cos 75^\circ + sin 75^\circ )(cos 10^\circ + sin 10^\circ)}{sin 15^\circ -i cos 15^\circ}is$

• A. 0
• B. $1/\sqrt{3}$
• C. i
• D. 1

Answer 1

Answer: (B)

Use of Trigonometric Identities
$ \cos x+\cos y=2 \cos \frac{(x+y)}{2} \cos \frac{(x-y)}{2}-(1) $
$ \sin x-\sin y=2 \cos \frac{(x+y)}{2} \sin \frac{(x-y)}{2}-(2) $
Calculation:
$ \begin{array}{l} \text { Given: } \\ \frac{\sin 75^{\circ}-\sin 15^{\circ}}{\cos 75^{\circ}+\cos 15^{\circ}} \end{array} $
From equation (1) and (2);
$ \begin{array}{l} =\frac{2 \cos \frac{75+15}{2} \cdot \sin \frac{75-15}{2}}{2 \cdot \cos \frac{75+15}{2} \cdot \cos \frac{75-15}{2}} \\ =\frac{\sin 30^{\circ}}{\cos 30^{\circ}} \\ =\tan 30=\frac{1}{\sqrt{3}} \end{array} $