Problem 986

First Solution: We have $\tan 65^{\circ}-2 \tan 40^{\circ}=\cot 25^{\circ}-2 \cot 50^{\circ}=\cot 25^{\circ}-\frac{\cot ^{2} 25^{\circ}-1}{\cot 25^{\circ}}=\frac{1}{\cot 25^{\circ}}=\tan 25^{\circ}$. Therefore, the answer is $25^{\circ}$. Second Solution: We have $\tan 65^{\circ}-2 \tan 40^{\circ}=\frac{1+\tan 20^{\circ}}{1-\tan 20^{\circ}}-\frac{4 \tan 20^{\circ}}{1-\tan ^{2} 20^{\circ}}=\frac{(1-\tan 20^{\circ})^{2}}{(1-\tan 20^{\circ})(1+\tan 20^{\circ})}=\tan (45^{\circ}-20^{\circ})=\tan 25^{\circ}$. Again, the answer is $25^{\circ}$.