A.  \(\large\boldsymbol{\frac{\cot\alpha\,\cdot\,\cot\beta\,-\,1 }{\cot\alpha\,+\,\cot\beta } }\)
B.  \(\large\boldsymbol{\frac{\cot\alpha\,\cdot\,\cot\beta\,-\,1 }{\cot\beta\,-\,\cot\alpha } }\)
C.  \(\large\boldsymbol{\frac{1-\cot\alpha\,\cdot\,\cot\beta }{\cot\alpha\,-\,\cot\beta } }\)
D.  \(\large\boldsymbol{\frac{\cot\alpha\,\cdot\,\cot\beta }{1\,+\,\cot\alpha\,\cdot\,\cot\beta } }\)
E.  \(\large\boldsymbol{\frac{\cot\alpha\,\cdot\,\cot\beta }{1\,-\,\cot\alpha\,\cdot\,\cot\beta } }\)
A    B    C    D    E

[ 5-1485 - op net sinds 21.4.2020-(E)-17.1.2025 ]

Translation in   E N G L I S H

See above

Oplossing - Solution

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