Question : If $\theta=135^{\circ} ; y=15^{\circ}$, what is the value of $2\cos(\theta)\sin(y)$?
Option 1: $\frac{\sqrt{3}-1}{2}$
Option 2: $\frac{1-\sqrt{3}}{2}$
Option 3: $2-\sqrt{3}$
Option 4: $\sqrt{3}-2$
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Correct Answer: $\frac{1-\sqrt{3}}{2}$
Solution : $\theta=135^{\circ}$ $y=15^{\circ}$ $2 \cos (\theta) \sin (y)$ = $2 \cos (135^{\circ}) \sin (15^{\circ})$ = $2 \cos (180^{\circ}-45^{\circ}) \sin (45^{\circ}-30^{\circ})$ = $2$ × $[-\cos45^{\circ}]$ × $[ \sin (45^{\circ})\cos(30^{\circ})-\cos (45^{\circ})\sin(30^{\circ})]$ = $2$ × $[ -\frac{1}{\sqrt2}]$ × $[ \frac{1}{\sqrt2}×\frac{\sqrt3}{2}-\frac{1}{\sqrt2}×\frac{1}{2}]$ = $2$ × $[-\frac{1}{\sqrt2}]$ × $[\frac{\sqrt3-1}{2\sqrt2}]$ = $\frac{1-\sqrt3}{2}$ Hence, the correct answer is $\frac{1-\sqrt3}{2}$.
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