a^2 9^ +b^2 27^ +c^2 63^ +d^2 81^ Let a, b, c and d be real numbers...

$$\left(a^2 \cot 9^{\circ}+d^2 \tan 9^{\circ}\right)+\left(b^2 \cot 27^{\circ}+c^2 \tan 27^{\circ}\right) ...(1)$$ Given $a+b+d=5$ Let $\quad a+d=k_1$ \begin{aligned} \therefore \quad b+c & =k_2 \\ \text { Let } \quad k_1+k_2 & =5 \\ E_1 & =a^2 \cot 9^{\circ}+d^2 \tan 9^{\circ} \\ E_1 & =a^2 \cot 9^{\circ}+\left(k_1-a\right)^2 \tan 9^{\circ} ...(2)\\ E_1 & =a^2\left(\cot 9^{\circ}+\tan 9^{\circ}\right)-a k_1 \tan 9^{\circ}+k_1^2 \tan 9^{\circ} \end{aligned} This is quadratic in $a$ whose minimum value is obtained at $x=\frac{-B}{2 A}$ i.e., $$a=\frac{2 k_1 \tan 9^{\circ}}{2\left(\cot 9^{\circ}+\tan 9^{\circ}\right)}=k_1 \sin ^2 9^{\circ}$$ Let \begin{aligned} E_2 & =b^2 \cot 27^{\circ}+c^2 \tan 27^{\circ} \\ b+c & =k_2 \\ E_2 & =b^2 \cot 27^{\circ}+\left(k^2-b\right)^2 \tan 27^{\circ} ...(3) \end{aligned} Make quadratic in $b$ $E_2 \mathrm{~min}$ is obtained $$b=k_2 \sin ^2 27^{\circ}$$ From Eqn. (2) \begin{aligned} \therefore \quad E_{1 \min } & =k_1^2 \sin ^4 9^{\circ} \cot 9^{\circ}+\left(k_1 \cos ^2 9\right)^2 \tan 9^{\circ} \\ & =k_1^2 \sin 9^{\circ} \cos 9^{\circ}\left(\sin ^2 9^{\circ}+\cos ^2 9^{\circ}\right) \\ & =k_1^2 \sin 9^{\circ} \cos 9^{\circ} \end{aligned} $$\begin{aligned} & \text { Similarly } E_2 \min =k_2^2 \sin 27^{\circ} \cos 27^{\circ} \\ & \therefore \quad E_{\text {min }}=E_{1 \text { min }}+E_{2 \text { min }} \\ & k_1^2 \sin 9^{\circ} \cos 9^{\circ}+k_2^2 \sin 27^{\circ} \cos 27^{\circ} \\ & =\frac{1}{2}\left[k_1^2 \sin 18^{\circ}+\left(5-k_1\right)^2 \sin 54^{\circ}\right] \\ & =\frac{1}{2}\left[k_1^2\left(\sin 18^{\circ}+\cos 36^{\circ}\right) 10 k_1 \cos 36^{\circ}+25 \cos 36^{\circ}\right] \\ & E_{\min }=\frac{-D}{4 A} \text { (quadratic in } k_1 \text { ) } \\ & =+\frac{25}{2}\left[\frac{\cos 36^{\circ} \sin 18^{\circ}}{\sin 18^{\circ}+\cos 36^{\circ}}\right] \\ & =\frac{25}{4 \sqrt{5}}=\frac{5 \sqrt{5}}{4}=\frac{\sqrt{125}}{4}=\frac{\sqrt{x}}{4} \\ & x+y=129 \\ & \end{aligned}$$