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Mathematics Question on Trigonometric Functions

The value of the expression 12(3sin⁡75∘−cos⁡ 75∘)\frac{1}{2} ( \sqrt{3} \sin 75^{\circ} - \cos \, 75^{\circ} ) is :

A

1

B

2

C

1/21 / \sqrt{2}

D

2/22 / \sqrt{2}

Answer

1/21 / \sqrt{2}

Explanation

Solution

Consider 12(3sin⁡75∘−cos⁡75∘)\frac{1}{2} \left(\sqrt{3} \sin75^{\circ} - \cos75^{\circ}\right) =32.sin⁡75∘−12cos⁡75∘= \frac{\sqrt{3}}{2} . \sin 75^{\circ} - \frac{1}{2} \cos75^{\circ} =sin⁡60∘.sin⁡75∘−cos⁡60∘.cos⁡75∘= \sin60^{\circ}. \sin75^{\circ} - \cos60^{\circ}. \cos75^{\circ} =−[cos⁡(75∘+60∘)]= - \left[\cos\left(75^{\circ}+ 60^{\circ}\right)\right] =−cos⁡135∘= - \cos135 ^{\circ} =−cos⁡(180∘−45∘)= - \cos\left(180^{\circ} - 45^{\circ}\right) =+12= + \frac{1}{\sqrt{2}}