wartosc wyrazenia

[kubass]

Wartosc wyrazenia \(\tg75^\circ - \tg15^\circ\) ile jest równa?

[Jerry]

\[\tg75^\circ - \tg15^\circ=\ctg15^\circ-\tg15^\circ=\\={\cos15^\circ\over\sin15^\circ}-{\sin15^\circ\over\cos15^\circ}={\cos^215^\circ-\sin^215^\circ\over\sin15^\circ\cos15^\circ}=\\={\cos(2\cdot15^\circ)\over{1\over2}\sin(2\cdot15^\circ)}={{\sqrt3\over2}\over{1\over2}\cdot{1\over2}}=2\sqrt3\]
Pozdrawiam

[eresh]

Wartosc wyrazenia \(\tg75^\circ - \tg15^\circ\) ile jest równa?

\(
\tg 75^{\circ}=\tg (30^{\circ}+45^{\circ})=\frac{\tg 30^{\circ}+\tg 45^{\circ}}{1-\tg 30^{\circ}\tg 45^{\circ}}=\frac{\frac{\sqrt{3}}{3}+1}{1-\frac{\sqrt{3}}{3}}=\frac{\sqrt{3}+3}{3}\cdot\frac{3}{3-\sqrt{3}}=\frac{\sqrt{3}+3}{3-\sqrt{3}}\\
\tg 15^{\circ}=\tg (45^{\circ}-30^{\circ})=\frac{\tg 45^{\circ}-\tg 30^{\circ}}{1+\tg 30^{\circ}\tg 45^{\circ}}=\frac{1-\frac{\sqrt{3}}{3}}{1+\frac{\sqrt{3}}{3}}=\frac{3-\sqrt{3}}{3}\cdot\frac{3}{3+\sqrt{3}}=\frac{3-\sqrt{3}}{3+\sqrt{3}}\\
\tg 75^{\circ}-\tg 15^{\circ}=\frac{\sqrt{3}+3}{3-\sqrt{3}}-\frac{3-\sqrt{3}}{3+\sqrt{3}}=\frac{(3+\sqrt{3})^2-(3-\sqrt{3})^2}{6}=\frac{12+6\sqrt{3}-12+6\sqrt{3}}{6}=2\sqrt{3}\)