Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\dpi{110}\frac {\cot \alpha\,+\,\cot \beta} {\tan \alpha\,+\,\tan\beta}$ is gelijk aan (uiteraard voor alle α, β die geen veelvoud zijn van 90°)	A. 1
B. tan α.tan β
C. cot α.cot β
D. tan²α + tan²β
E. cot²α + cot²β

$https://latex.codecogs.com/png.image?\dpi{110}\frac {\cot \alpha\,+\,\cot \beta} {\tan \alpha\,+\,\tan\beta}$ is equal to	A. 1
B. tanα.tanβ
C. tanα.tanβ
D. tan²α + tan²β
E. cot²α + cot²β

1^ste manier :
\(\frac{\cot\alpha\,+\,\cot\beta}{\tan\alpha\,+\,\tan\beta}=\frac{\frac{1}{\tan\alpha}+\frac{1}{\tan\beta}}{\tan\alpha\,+\,\tan\beta}=\frac{\frac{\tan\beta\,+\,\tan\alpha}{\tan\alpha\,.\,\tan\beta}}{\tan\alpha\,+\,\tan\beta}\\=\frac{\tan\beta\,+\,\tan\alpha}{\tan\alpha\,.\,\tan\beta}.\frac{1}{\tan\alpha\,+\,\tan\beta}=\frac{1}{\tan\alpha\,.\,\tan\beta}=\cot\alpha.\cot\beta\)
2^de manier :
\(\large\frac{\cot\alpha\,+\,\cot\beta}{\tan\alpha\,+\,\tan\beta}=\frac{\cot\alpha\,+\,\cot\beta}{\frac{1}{\cot\alpha}+\frac{1}{\cot\beta}}=\frac{\cot\alpha\,+\,\cot\beta}{\frac{\cot\beta\,+\,\cot\alpha}{\cot\alpha\,.\,\cot\beta}}\\=(\cot\alpha\,+\,\cot\beta).\frac{\cot\alpha\,.\,\cot\beta}{\cot\beta\,+\,\cot\alpha}=\cot\alpha\,.\,\cot\beta\)
3^de manier :
\(\large\frac{\cot\alpha\,+\,\cot\beta}{\tan\alpha\,+\,\tan\beta}=\frac{\frac{\cos\alpha}{\sin\alpha}+\frac{\cos\beta}{\sin\beta}}{\frac{\sin\alpha}{\cos\alpha}+\frac{\sin\beta}{\cos\beta}}=\frac{\frac{\cos\alpha.\sin\beta+\cos\beta.\sin\alpha}{\sin\alpha.\sin\beta}}{\frac{\sin\alpha.\cos\beta+\sin\beta.\cos\alpha}{\cos\alpha.\cos\beta}}\\=\frac{\cos\alpha.\sin\beta+\cos\beta.\sin\alpha}{\sin\alpha.\sin\beta}.\frac{\cos\alpha.\cos\beta}{\sin\alpha.\cos\beta+\sin\beta.\cos\alpha}\\=\frac{\cos\alpha.\cos\beta}{\sin\alpha.\sin\beta}=\frac{\cos\alpha}{\sin\alpha}.\frac{\cos\beta}{\sin\beta}=\cot\alpha.\cot\beta\)