1.
[tex]cos30^{o} = \frac{\sqrt{3}}{2}\\tg30^{o} = \frac{\sqrt{3}}{3}\\tg60^{o} = \sqrt{3}\\sin^{2}\alpha + cos^{2}\alpha = 1[/tex]
[tex]\frac{4cos^{2}30^{o}-tg30^{o}\cdot tg60^{o}}{sin^227^{o}+sin^{2}63^{o}} = \frac{4\cdot(\frac{\sqrt{3}}{2})^{2}-\frac{\sqrt{3}}{3}\cdot\sqrt{3}}{sin^{2}27^{o}+sin^{2}(90^{o}-27^{o})} = \frac{4\cdot\frac{3}{4}-\frac{3}{3}}{sin^{2}27^{o}+cos^{2}27^{o}} = \frac{3-1}{1} = 2[/tex]
2.
[tex]tg\alpha = \frac{5}{12}\\\\\frac{sin\alpha}{cos\alpha} = tg\alpha = \frac{5}{12} \ \ \ |()^{2}\\\\\frac{sin^{2}\alpha}{cos^{2}\alpha} = \frac{25}{144}[/tex]
sin²α + cos²α = 1 ⇒ cos²α = 1 - sin²α
[tex]\frac{sin^{2}\alpha}{1-sin^{2}\alpha} = \frac{25}{144}\\\\144sin^{2}\alpha = 25(1-sin^{2}\alpha)\\\\144sin^{2}\alpha = 25 - 25sin^{2}\alpha\\\\144sin^{2}\alpha + 25sin^{2}\alpha = 25\\\\169sin^{2}\alpha = 25 \ \ \ /:168\\\\sin^{2}\alpha = \frac{25}{169}\\\\sin\alpha = \sqrt{\frac{25}{169}}\\\\\boxed{sin\alpha = \frac{5}{13}}[/tex]
[tex]cos^{2}\alpha = 1-sin^{2}\alpha\\\\cos^{2}\alpha = 1-(\frac{5}{13})^{2} = \frac{169}{169} - \frac{25}{169} = \frac{144}{169}\\\\cos\alpha = \sqrt{\frac{144}{169}}\\\\\boxed{cos\alpha = \frac{12}{13}}[/tex]
3.
[tex]tg\alpha = 4\\\\\frac{sin\alpha}{cos\alpha} = 4\\\\sin\alpha = 4cos\alpha[/tex]
[tex]\frac{3sin\alpha - 5cos\alpha}{sin\alpha + 8cos\alpha} = \frac{3\cdot4cos\alpha - 5cos\alpha}{4cos\alpha+8cos\alpha}=\frac{12cos\alpha - 5cos\alpha}{12cos\alpha} = \frac{7cos\alpha}{12cos\alpha} = \frac{7}{121}[/tex]
