Вычислите [latex]\tan9^\circ-\tan63^\circ+\tan81^\circ-\tan27^\circ;\\ \cos^235^\circ+\cos^225^\circ-\cos^25^\circ.[/latex]

[latex]\tan9а-\tan63а+\tan81а-\tan27а=\\ =\tan9а-\tan27а+\tan(90а-9а)-\tan(90а-27а)=\\ =\tan9а-\tan27а+\cot9а-\cot27а=\\ = \frac{\sin(9а-27а)}{\cos9а\cos27а}- \frac{\sin(9а-27а)}{\sin9а\sin27а}= \frac{\sin18а(\cos9а\cos27а-\sin9а\sin27а)}{\sin9а\sin27а\cos9а\cos27а}=\\ = \frac{2\sin9а\cos9а\cos(9а+27а)}{\sin9а\cos9а\sin27а\cos27а} = \frac{2\cos36а}{0.5\sin54а} = \frac{4\cos36а}{\sin(90а-36а)} = \frac{4\cos36а}{\cos36а}=4 [/latex] [latex]\cos^235а+\cos^225а-\cos^25а= \frac{1+\cos70а}{2}+ \frac{1+\cos50а}{2} - \frac{1+\cos10а}{2} = \\ = \frac{1}{2}+ \frac{\cos(90а-20а)+1+\cos(90а-40а)-\cos(90а-80а)}{2} = \\ = \frac{1}{2} + \frac{\sin20а+\sin40а-\sin80а}{2}= \frac{1}{2} + \frac{\sin20а+2\sin20а\cos20а-2\sin40а\cos40а}{2}= \\ = \frac{1}{2}+ \frac{\sin20а+2\sin20а\cos20а-4\sin20а\cos20а\cos40а}{2}=\\ = \frac{1}{2}+ \frac{\sin20а(1+2\cos20а-4\cos20а\cos40а)}{2} = [/latex] [latex] =\frac{1}{2} + \frac{\sin20а(1+2\cos20а-4\cdot \frac{\cos(40а-20а)+\cos(40а+20а)}{2} }{2}=\\ = \frac{1}{2}+ \frac{\sin20а(1+2\cos20а-2\cos20а-2\cos60а)}{2} = \frac{1}{2} [/latex]