​If $$\sqrt{3}$$ sinθ – cosθ= 0 (θ is an acute angle), then the value of $$\text{cos}^3$$θ – $$\sqrt{3}$$ $$\text{sin}^3$$θ will be: ​

Correct option is A

Given:

​$$\sqrt 3$$​s​in θ - cos θ = 0, and θ is an acute angle.

Solution:

​$$\sqrt 3$$​​s​in θ - cos θ = 0,

$$\sqrt 3$$​ sin θ = cos θ

​$$\frac{(\sqrt 3 sin θ) }{ cos θ}$$​ = 1

tan θ = $$\frac{1}{\sqrt 3}$$​​

θ is an acute angle (0° < θ < 90°), we know that θ = 30°.

cos³ θ - $$\sqrt 3$$​​ sin³ θ

=cos³30°- $$\sqrt 3$$​​ sin³ 30°

$$=\left( \frac{\sqrt{3}}{2} \right)^3 - \sqrt{3} \left( \frac{1}{2} \right)^3 \\= \frac{3\sqrt{3}}{8} - \sqrt{3} \times \frac{1}{8} \\= \frac{3\sqrt{3}}{8} - \frac{\sqrt{3}}{8} \\= \frac{3\sqrt{3} - \sqrt{3}}{8} \\= \frac{2\sqrt{3}}{8} \\= \frac{\sqrt{3}}{4}$$​ 

Option (a) is right.