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81.
$\sin(4\alpha)=4\sin\alpha \cos\alpha -8\sin^3\alpha \cos\alpha$
82.
$\sin \left(5\alpha\right) = 5\sin\alpha - 20\sin^3\alpha+16\sin^5\alpha$
83.
$\cos3\alpha=4\cos^3\alpha-3\cos\alpha$
84.
$\cos3\alpha=\cos^3\alpha-3\cos\alpha\sin^2\alpha$
85.
$\cos4\alpha = 8\cos^4\alpha-8\cos^2\alpha+1$
86.
$\cos5\alpha=16\cos^5\alpha-20\cos^3\alpha+5\cos\alpha$
87.
$\tan3\alpha=\dfrac{3\tan \alpha-\tan^3\alpha}{1-3\tan^2\alpha}$
88.
$\cot3\alpha=\dfrac{\cot^3\alpha-3\cot\alpha}{3\cot^2\alpha-1}$
89.
Sum of sines
$\sin \alpha + \sin \beta = 2 \cdot \sin\dfrac{\alpha+\beta}{2}\cdot \cos \dfrac{\alpha-\beta}{2}$
90.
Difference of sines
$\sin \alpha - \sin \beta = 2\cdot \sin\dfrac{\alpha - \beta}{2} \cdot \cos \dfrac{\alpha+\beta}{2}$
91.
Sum of cosines
$\cos\alpha + \cos\beta = 2 \cdot \cos \dfrac{\alpha+\beta}{2} \cdot \cos \dfrac{\alpha-\beta}{2}$
92.
Difference of cosines
$\cos\alpha - \cos\beta = -2 \cdot \sin \dfrac{\alpha+\beta}{2} \cdot \sin \dfrac{\alpha-\beta}{2}$
93.
Sum of tangents
$\tan \alpha + \tan \beta = \dfrac{\sin(\alpha+\beta)}{\cos\alpha \cdot \cos \beta}$
94.
Difference of tangents
$\tan \alpha - \tan \beta = \dfrac{\sin(\alpha-\beta)}{\cos\alpha \cdot \cos \beta}$
95.
Sum of cotangents
$\cot \alpha + \cot \beta = \dfrac{\sin(\beta + \alpha)}{\sin\alpha \cdot \sin\beta}$
96.
Difference of cotangents
$\cot \alpha - \cot \beta = \dfrac{\sin(\beta - \alpha)}{\sin\alpha \cdot \sin\beta}$
97.
$\cos\alpha + \sin\alpha = \sqrt{2} \cos\left( \frac{\pi}{4}-\alpha\right)$
98.
$\cos\alpha - \sin\alpha = \sqrt{2} \sin\left( \frac{\pi}{4}-\alpha\right)$
99.
Product of sines
$\sin\alpha \cdot \sin \beta = \dfrac{1}{2} \left[ \cos(\alpha-\beta) - \cos(\alpha + \beta) \right]$
100.
Product of cosines
$\cos\alpha \cdot \cos \beta = \dfrac{1}{2} \left[ \cos(\alpha-\beta) + \cos(\alpha + \beta) \right]$