The value of 9^ ^ - 27^ ^ - 63^ ^ + 81^ ^ is equal to

$$\begin{aligned} & \tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ} \\ & =\tan 9^{\circ}-\tan 27^{\circ}-\tan \left(90^{\circ}-27^{\circ}\right)+\tan \left(90^{\circ}-9^{\circ}\right)\end{aligned}$$$$\begin{aligned} & =\tan 9^{\circ}-\tan 27^{\circ}-\cot 27^{\circ}+\cot 9^{\circ} \\ & =\tan 9^{\circ}+\cot 9^{\circ}-\left(\tan 27^{\circ}+\cot 27^{\circ}\right)\end{aligned}$$ We know, $$\begin{aligned} & \tan \theta+\cot \theta=\frac{1}{\sin \theta \cos \theta}=\frac{2}{2 \sin \theta \cos \theta}=\frac{2}{\sin 2 \theta} \\ & \therefore \tan 9^{\circ}+\cot 9^{\circ}-\left(\tan 27^{\circ}+\cot 27^{\circ}\right)\end{aligned}$$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}}=2 \left(\frac{\sin 54^{\circ}-\sin 18^{\circ}}{\sin 18^{\circ} \sin 54^{\circ}}\right) We also know, $\sin C-\sin$D$$\begin{aligned}& =2 \cos \frac{(\mathrm{C}+\mathrm{D})}{2} \sin \left(\frac{\mathrm{C}-\mathrm{D}}{2}\right) .\\ & \therefore 2\left(\frac{\sin 54^{\circ}-\sin 18^{\circ}}{\sin 18^{\circ} \sin 54^{\circ}}\right)=2 \cdot 2 \cdot \frac{\cos 36^{\circ} \sin 18^{\circ}}{\sin 18^{\circ} \sin 54^{\circ}}\end{aligned}$$=$2 \cdot 2 \cdot$\frac{\sin 54^{\circ} \cdot \sin 18^{\circ}}{\sin 18^{\circ} \cdot \sin 54^{\circ}}=$4