Problem Statement: Find the value of $\cos^3\theta + \sec^3\theta$ given that $\cos\theta + \sec\theta = \sqrt{3}$.

Solving for $\cos^3\theta + \sec^3\theta$

We are given the equation:
$ \cos\theta + \sec\theta = \sqrt{3} $ We need to find the value of $\cos^3\theta + \sec^3\theta$.

Let $a = \cos\theta$ and $b = \sec\theta$. The expression we need to find is $a^3 + b^3$. We know the algebraic identity: $ a^3 + b^3 = (a+b)^3 - 3ab(a+b) $ In this case, $a = \cos\theta$ and $b = \sec\theta$. The product $ab$ is: $ ab = \cos\theta \cdot \sec\theta = \cos\theta \cdot \frac{1}{\cos\theta} = 1 $ Now substitute $a+b = \sqrt{3}$ and $ab = 1$ into the identity: $ \cos^3\theta + \sec^3\theta = (\cos\theta + \sec\theta)^3 - 3(\cos\theta \sec\theta)(\cos\theta + \sec\theta) $ $ \cos^3\theta + \sec^3\theta = (\sqrt{3})^3 - 3(1)(\sqrt{3}) $ Calculate the terms: $ (\sqrt{3})^3 = \sqrt{3} \times \sqrt{3} \times \sqrt{3} = 3\sqrt{3} $ $ 3(1)(\sqrt{3}) = 3\sqrt{3} $ Substitute these values back into the equation: $ \cos^3\theta + \sec^3\theta = 3\sqrt{3} - 3\sqrt{3} $ $ \cos^3\theta + \sec^3\theta = 0 $

Therefore, the value of $\cos^3\theta + \sec^3\theta$ is 0.

Conclusion: The correct option is 0.