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41.
Difference of cosines
$\cos\alpha - \cos\beta = -2 \cdot \sin \dfrac{\alpha+\beta}{2} \cdot \sin \dfrac{\alpha-\beta}{2}$
42.
Sum of tangents
$\tan \alpha + \tan \beta = \dfrac{\sin(\alpha+\beta)}{\cos\alpha \cdot \cos \beta}$
43.
Difference of tangents
$\tan \alpha - \tan \beta = \dfrac{\sin(\alpha-\beta)}{\cos\alpha \cdot \cos \beta}$
44.
Sum of cotangents
$\cot \alpha + \cot \beta = \dfrac{\sin(\beta + \alpha)}{\sin\alpha \cdot \sin\beta}$
45.
Difference of cotangents
$\cot \alpha - \cot \beta = \dfrac{\sin(\beta - \alpha)}{\sin\alpha \cdot \sin\beta}$
48.
Product of sines
$\sin\alpha \cdot \sin \beta = \dfrac{1}{2} \left[ \cos(\alpha-\beta) - \cos(\alpha + \beta) \right]$
49.
Product of cosines
$\cos\alpha \cdot \cos \beta = \dfrac{1}{2} \left[ \cos(\alpha-\beta) + \cos(\alpha + \beta) \right]$
50.
Product of sine and cosine
$\sin\alpha \cdot \cos \beta = \dfrac{1}{2} \left[ \sin(\alpha-\beta) + \sin(\alpha + \beta) \right]$
51.
Product of tangents
$\tan\alpha \cdot \tan\beta = \dfrac{\tan \alpha + \tan \beta}{\cot \alpha + \cot \beta}$
52.
Product of cotangents
$\cot\alpha \cdot \cot\beta = \dfrac{\cot \alpha + \cot \beta}{\tan \alpha + \tan \beta}$
53.
Product of tangent and cotangents
$\tan\alpha \cdot \cot\beta = \dfrac{\tan\alpha + \cot\beta}{\cot\alpha+\tan\beta}$