Question

The value of $\frac{(cos20^{\circ }+sin20^{\circ })(cos75^{\circ }+sin75^{\circ })(cos10^{\circ }+sin10^{\circ })}{sin15^{\circ }-icos15^{\circ }}is$

• 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);
$\begin{array}{l}=\frac{2\cos \frac{75+15}{2}\cdot \sin \frac{75-15}{2}}{2\cdot \cos \frac{75+15}{2}\cdot \cos \frac{75-15}{2}}\\ =\frac{\sin 30^{\circ }}{\cos 30^{\circ }}\\ =\tan 30=\frac{1}{\sqrt{3}}\end{array}$