Trigonometry

Do you want it simplified? It already is. What are you looking for in this expression?

$\frac{(cot\alpha cot\beta -1)}{(cot\alpha +cot\beta )}\\ =\frac{cos\alpha cos\beta-sin\alpha sin\beta}{sin\alpha sin\beta}\div \left(\frac{cos\alpha}{sin\alpha}+\frac{cos\beta}{sin\beta}\right)\\ =\frac{cos\alpha cos\beta-sin\alpha sin\beta}{sin\alpha sin\beta}\div \frac{2cos\alpha sin\beta}{sin\alpha sin\beta}\\ =\frac{cos\alpha cos\beta-sin\alpha sin\beta}{2cos\alpha sin\beta}\\ =\frac{cos\alpha cos\beta-sin\alpha sin\beta}{2cos\alpha sin\beta}\\ =\frac{cos(\alpha+\beta)}{sin(\alpha+\beta)}\\ =cot(\alpha+\beta)$