JEE 2023 — Mathematics PYQ

$$(\tan 9^{\circ }+\cot 9^{\circ })-(\tan 27^{\circ }+\cot 27^{\circ })$$
$$\Rightarrow \left(\frac{{\sin }^2{9}^{\circ }+{\cos }^2{9}^{\circ }}{\sin 9^{\circ }\cos 9^{\circ }}\right)-\left(\frac{{\sin }^{2}27^{\circ }+{\cos }^{2}27^{\circ }}{\sin 27^{\circ }\cos 27^{\circ }}\right)$$
$$=\frac{2}{2\sin 9^{\circ }\cos 9^{\circ }}-\frac{2}{2\sin 27^{\circ }\cos 27^{\circ }}$$
$$\Rightarrow \frac{2}{\sin 18^{\circ }}-\frac{2}{\sin 54^{\circ }}[\because \sin 2A=2\sin A\cos A]$$
$$\Rightarrow \frac{2}{\frac{\sqrt{5}-1}{4}}-\frac{2}{\frac{\sqrt{5}+1}{4}}$$
$$\because \sin 18^{\circ }=\frac{\sqrt{5}-1}{4}$$
$$\because \sin 54^{\circ }=\frac{\sqrt{5}+1}{4}$$