D=5
![a_{n}=a_{1} +d(n-1)](https://tex.z-dn.net/?f=+a_%7Bn%7D%3Da_%7B1%7D++%2Bd%28n-1%29)
89=-6+5(n-1) ⇒ 5n=100 ⇒ n=20 (номер члена прогрессии)
411) Сначала преобразуем выражение по формулам приведения
![ctg(\frac{3 \pi}{2} + \alpha)=-tg\alpha](https://tex.z-dn.net/?f=ctg%28%5Cfrac%7B3+%5Cpi%7D%7B2%7D+%2B+%5Calpha%29%3D-tg%5Calpha)
Тангенс и котангенс взаимно обратные величины, из этого следует что:
![tg\alpha=\frac{1}{ctg\alpha}=\frac{11}{10}](https://tex.z-dn.net/?f=tg%5Calpha%3D%5Cfrac%7B1%7D%7Bctg%5Calpha%7D%3D%5Cfrac%7B11%7D%7B10%7D)
Конечный ответ: ![ctg(\frac{3 \pi}{2} + \alpha)=-tg\alpha=-\frac{11}{10}](https://tex.z-dn.net/?f=ctg%28%5Cfrac%7B3+%5Cpi%7D%7B2%7D+%2B+%5Calpha%29%3D-tg%5Calpha%3D-%5Cfrac%7B11%7D%7B10%7D)
413)![2tg1095^{\circ}+ctg975^{\circ}-tg(-195^{\circ}) = 2tg1095^{\circ}+ctg975^{\circ}+tg195^{\circ}=2tg(3 \cdot 360^{\circ}+15^{\circ}) + ctg(2 \cdot 360^{\circ}+270^{\circ}-15^{\circ})+tg(180^{\circ}+15^{\circ})=2tg15^{\circ}+tg15^{\circ}+tg15^{\circ}=\\=4tg15^{\circ}=4(2-\sqrt3)](https://tex.z-dn.net/?f=2tg1095%5E%7B%5Ccirc%7D%2Bctg975%5E%7B%5Ccirc%7D-tg%28-195%5E%7B%5Ccirc%7D%29+%3D+2tg1095%5E%7B%5Ccirc%7D%2Bctg975%5E%7B%5Ccirc%7D%2Btg195%5E%7B%5Ccirc%7D%3D2tg%283+%5Ccdot+360%5E%7B%5Ccirc%7D%2B15%5E%7B%5Ccirc%7D%29+%2B+ctg%282+%5Ccdot+360%5E%7B%5Ccirc%7D%2B270%5E%7B%5Ccirc%7D-15%5E%7B%5Ccirc%7D%29%2Btg%28180%5E%7B%5Ccirc%7D%2B15%5E%7B%5Ccirc%7D%29%3D2tg15%5E%7B%5Ccirc%7D%2Btg15%5E%7B%5Ccirc%7D%2Btg15%5E%7B%5Ccirc%7D%3D%5C%5C%3D4tg15%5E%7B%5Ccirc%7D%3D4%282-%5Csqrt3%29)
![(5a^m+3b^m)(5a^m-3b^m)=25a^{2m}-9b^{2m}](https://tex.z-dn.net/?f=%285a%5Em%2B3b%5Em%29%285a%5Em-3b%5Em%29%3D25a%5E%7B2m%7D-9b%5E%7B2m%7D)
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