Sprowadz wyrazenie do najprostszej postaci

[truskawka89]

Sprowadz wyrazenie do najprostszej postaci:\(\displaystyle{ W= \frac{tg\alpha + tg\beta}{ctg\alpha+ctg\beta}}\) a nastepnie oblicz jego wartosc dla takich katow ze, \(\displaystyle{ ctg\alpha= 4 \beta=180 stopni - }\)[/latex]

[natkoza]

\(\displaystyle{ W=\frac{tg\alpha+tg\beta}{ctg\alpha+ctg\beta}=\frac{\frac{sin\alpha}{cos\alpha}+\frac{sin\beta}{cos\beta}}{\frac{cos\alpha}{sin\alpha}+\frac{cos\beta}{sin\beta}}=\frac{\frac{sin\alpha cos\beta+sin\beta cos\alpha}{cos\alpha cos\beta}}{\frac{cos\alpha sin\beta+sin\alpha cos\beta}{sin\alpha sin\beta}}=\\
\frac{sin\alpha cos\beta+sin\beta cos\alpha}{cos\alpha cos\beta}\cdot \frac{sin\alpha sin\beta}{cos\alpha sin\beta+sin\alpha cos\beta}=\frac{sin\alpha sin\beta}{cos\alpha cos\beta}}\)

[mat1989]

\(\displaystyle{ =tg\alpha tg\beta}\)
\(\displaystyle{ tg =\frac{1}{ctg }}\)
\(\displaystyle{ tg \beta=tg 180-\alpha=-tg\alpha}\)