$\displaystyle \begin{array}{{>{\displaystyle}l}} 1.\ Ta\ có\ cos2a=2cos^{2} a-1=1-2sin^{2} a=\frac{1}{4}\\ \Rightarrow cos^{2} a=\frac{5}{8} ;\ sina=\pm \frac{\sqrt{6}}{4}\\ sin2acosa=2sinacos^{2} a=\pm 2.\frac{\sqrt{6}}{4} .\frac{5}{8} =\pm \frac{5\sqrt{6}}{16}\\ 2.\ \frac{cosx}{sinx-cosx} +\frac{sinx}{sinx+cosx}\\ =\frac{cosxsinx+cos^{2} x+sin^{2} x-cosxsinx}{sin^{2} x-cos^{2} x}\\ =\frac{1}{sin^{2} x-cos^{2} x} =\frac{\frac{1}{sin^{2} x}}{1-\frac{cos^{2} x}{sin^{2} x}} =\frac{1+cot^{2} x}{1-cot^{2} x}\\ 3.\ cot15^{o} =tan75^{o} =tan\ \left( 45^{o} +30^{o}\right)\\ =\frac{tan45^{o} +tan30^{o}}{1-tan45^{o} tan30^{o}} =2+\sqrt{3}\\ 4.\ A=8sin^{3} 18^{o} +8sin^{2} 18^{o} =1\\ Chứng\ minh:\\ Có\ sin18^{o} =cos72^{o} =2cos^{2} 36^{o} -1=2\left( 1-2sin^{2} 18^{o}\right)^{2} -1\\ A-1=0\\ \Leftrightarrow 8sin^{4} 18^{o} -8sin^{2} 18^{0} -sin^{2} 18+1=0\\ \Leftrightarrow \left( sin18^{o} -1\right)\left( 8sin^{3} 18^{o} +8sin^{2} 18^{0} -1\right) =0 \end{array}$