Tìm x để:

\(-\frac{\sqrt{x}+6}{\sqrt{x}-2}< 3\)

Rút gọn: \(\frac{1+\frac{\sqrt{3}}{2}}{1+\sqrt{1+\frac{\sqrt{3}}{2}}}+\frac{1-\frac{\sqrt{3}}{2}}{1-\sqrt{1-\frac{\sqrt{3}}{2}}}\)

Rút gọn: \(\frac{\sqrt{12-2\sqrt{35}}+\sqrt{8-2\sqrt{15}}+\sqrt{5-2\sqrt{6}}}{\sqrt{12+2\sqrt{35}}-\sqrt{8+2\sqrt{15}}+\sqrt{5-2\sqrt{6}}}\)

Rút gọn: \(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2}+\sqrt{3}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2}-\sqrt{3}}\)

Giải PT:

a) \(\sqrt{x+1}-\sqrt{x-2}=1\)

b) \(x^2-\sqrt{x^2-2}=4\)

c) \(\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}=4-2x-x^2\)

Cho \(\tan\alpha=\frac{3}{5}\), hãy tính giá trị của:

a) \(M=\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)

b) \(N=\frac{\sin\alpha\cos\alpha}{\sin^2\alpha-\cos^2\alpha}\)

c) \(P=\frac{\sin^3\alpha+\cos^3\alpha}{2\sin\alpha\cos^2\alpha+\cos\alpha\sin^2\alpha}\)

CM các hệ thức sau:

a) \(1+\tan^2\alpha=\frac{1}{\cos^2\alpha}\)
b) \(1+\cot^2\alpha=\frac{1}{\sin^2\alpha}\)
c) \(\cot^2\alpha-\cos^2\alpha=\cot^2\alpha.\cos^2\alpha\)
d) \(\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}\)