If the value of \[ \frac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ} = \frac{a\sqrt{5} - b}{c}, \] where \(a, b, c\) are natural numbers and \(\text{gcd}(a, c) = 1\), then \(a + b + c\) is equal to:
- 50
- 40
- 52
- 54
The Correct Option is C
Solution and Explanation
\( \dfrac{3\left(\dfrac{\sqrt{5}+1}{4}\right) + 5\left(\dfrac{\sqrt{5}-1}{4}\right)}{5\left(\dfrac{\sqrt{5}+1}{4}\right) - 3\left(\dfrac{\sqrt{5}-1}{4}\right)} = \dfrac{8\sqrt{5}-2}{2\sqrt{5}+8} \)
\( = \dfrac{4\sqrt{5}-1}{\sqrt{5}+4} \times \dfrac{\sqrt{5}-4}{\sqrt{5}-4} \)
\( = \dfrac{20 - 16\sqrt{5} - \sqrt{5} + 4}{-11} \)
\( = \dfrac{17\sqrt{5}-24}{11} \Rightarrow a = 17, \, b = 27, \, c = 11 \)
\( a + b + c = 52 \)
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