If the value of $$\frac{3\cos 36° + 5\sin 18°}{5\cos 36° - 3\sin 18°}$$ is $$\frac{a\sqrt{5} - b}{c}$$, where $$a, b, c$$ are natural numbers and $$\gcd(a, c) = 1$$, then $$a + b + c$$ is equal to :

We need to evaluate $$\frac{3\cos 36° + 5\sin 18°}{5\cos 36° - 3\sin 18°}$$.

Using the known values:

$$\cos 36° = \frac{\sqrt{5}+1}{4}, \quad \sin 18° = \frac{\sqrt{5}-1}{4}$$

$$3\cos 36° + 5\sin 18° = 3 \cdot \frac{\sqrt{5}+1}{4} + 5 \cdot \frac{\sqrt{5}-1}{4}$$

$$= \frac{3\sqrt{5}+3+5\sqrt{5}-5}{4} = \frac{8\sqrt{5}-2}{4} = \frac{4\sqrt{5}-1}{2}$$

$$5\cos 36° - 3\sin 18° = 5 \cdot \frac{\sqrt{5}+1}{4} - 3 \cdot \frac{\sqrt{5}-1}{4}$$

$$= \frac{5\sqrt{5}+5-3\sqrt{5}+3}{4} = \frac{2\sqrt{5}+8}{4} = \frac{\sqrt{5}+4}{2}$$

$$\frac{4\sqrt{5}-1}{2} \div \frac{\sqrt{5}+4}{2} = \frac{4\sqrt{5}-1}{\sqrt{5}+4}$$

Rationalizing by multiplying by $$\frac{\sqrt{5}-4}{\sqrt{5}-4}$$:

$$= \frac{(4\sqrt{5}-1)(\sqrt{5}-4)}{(\sqrt{5}+4)(\sqrt{5}-4)} = \frac{20-16\sqrt{5}-\sqrt{5}+4}{5-16} = \frac{24-17\sqrt{5}}{-11} = \frac{17\sqrt{5}-24}{11}$$

So the expression equals $$\frac{17\sqrt{5} - 24}{11}$$, where $$a = 17, b = 24, c = 11$$.

Checking: $$\gcd(17, 11) = 1$$. ✓

Therefore $$a + b + c = 17 + 24 + 11 = 52$$.

The correct answer is Option 2: 52.