If A = 10°, what is the value of: 12 3A + 5 ( 5A - 5^ )9 9A2 - 4 ( 5A + 10^ )?

Answer: Option A

$$\begin{array}{rl}& A=10^{\circ }\\ & \frac{12\sin 3A+5\cos \left(5A-5^{\circ }\right)}{9\sin \frac{9A}{2}-4\cos \left(5A+{10}^{\circ }\right)}\\ & =\frac{12\sin 3\times {10}^{\circ }+5\cos \left(5\times {10}^{\circ }-5^{\circ }\right)}{9\sin \frac{9\times {10}^{\circ }}{2}-4\cos \left(5\times {10}^{\circ }+{10}^{\circ }\right)}\\ & =\frac{12\sin {30}^{\circ }+5\cos \left({50}^{\circ }-5^{\circ }\right)}{9\sin \frac{{90}^{\circ }}{2}-4\cos \left({50}^{\circ }+{10}^{\circ }\right)}\\ & =\frac{12\sin {30}^{\circ }+5\cos {45}^{\circ }}{9\sin {45}^{\circ }-4\cos {60}^{\circ }}\\ & =\frac{12\times \frac{1}{2}+5\times \frac{1}{\sqrt{2}}}{9\times \frac{1}{\sqrt{2}}-4\times \frac{1}{2}}\\ & =\frac{6+\frac{5}{\sqrt{2}}}{\frac{9}{\sqrt{2}}-2}\\ & =\frac{6\sqrt{2}+5}{9-2\sqrt{2}}\end{array}$$