দেওয়া আছে, $\cos \theta = \frac{1}{2}(a + \frac{1}{a})$। আমরা জানি, $\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$। $$ \begin{aligned} \cos 3\theta &= 4 \left[ \frac{1}{2} \left( a + \frac{1}{a} \right) \right]^3 - 3 \left[ \frac{1}{2} \left( a + \frac{1}{a} \right) \right] \\ &= 4 \cdot \frac{1}{8} \left( a + \frac{1}{a} \right)^3 - \frac{3}{2} \left( a + \frac{1}{a} \right) \\ &= \frac{1}{2} \left[ \left( a + \frac{1}{a} \right)^3 - 3 \left( a + \frac{1}{a} \right) \right] \\ &= \frac{1}{2} \left[ a^3 + \frac{1}{a^3} + 3(a + \frac{1}{a}) - 3(a + \frac{1}{a}) \right] \\ &= \frac{1}{2} \left( a^3 + \frac{1}{a^3} \right) \end{aligned} $$ সুতরাং, সঠিক উত্তর $\frac{1}{2}\left(a^{3}+\frac{1}{a^{3}}\right)$।