Показать, что tan (52,5 °) = sqrt6 - sqrt3 - sqrt2 + 2?

# Rarrtan75 ° = загар (45 + 30) #

# = (Tan45 + tan30) / # (1-tan45 * tan30)

# = (1+ (1 / SQRT (3))) / (1- (1 / SQRT (3)) #

# = (SQRT (3) + 1) / (SQRT (3) -1) = 2 + SQRT (3) #

# Rarrtan52.5 = кроватка (90-37.5) = cot37.5 #

# Rarrcot37.5 = 1 / (тангенс (75/2)) #

# Rarrtanx = (2tan (х / 2)) / (1-тан ^ 2 (х / 2)) #

# Rarrtanx-Tanx * загар ^ 2 (х / 2) = 2tan (х / 2) #

# Rarrtanx * загар ^ 2 (х / 2) + 2tan (х / 2) -tanx = 0 #

Это квадратично в #tan (х / 2) # Так, #rarrtan (х / 2) = (- 2 + SQRT (2 ^ 2-4 * Tanx * (- Tanx))) / (2 * Tanx) #

#rarrtan (х / 2) = (- 2 + SQRT (4 (1 + загар ^ 2x))) / (2 * Tanx) #

#rarrtan (х / 2) = (- 1 + SQRT (1 + загар ^ 2x)) / Tanx #

Ввод # Х = 75 # мы получаем

#rarrtan (75/2) = (- 1 + SQRT (1 + загар ^ 2 (75))) / (tan75) #

#rarrtan (75/2) = (- 1 + SQRT (1+ (2 + SQRT (3)) ^ 2)) / (2 + SQRT (3)) #

#rarrtan (75/2) = (- 1 + SQRT (1 + 4 + 4sqrt (3) + 3)) / (2 + SQRT (3)) #

#rarrtan (75/2) = (- 1 + SQRT (8 + 4sqrt (3))) / (2 + SQRT (3)) #

# Rarr1 / тангенс (75/2) = (2 + SQRT (3)) / (2 * SQRT (2 + SQRT (3)) - 1) * (2 * SQRT (2 + SQRT (3)) + 1) / (2 * SQRT (2 + SQRT (3) + 1) #

# Rarrcot37.5 = (2 * (2 * SQRT (2 + SQRT (3)) + 1) SQRT (3) * (2 * SQRT (2 + SQRT (3)) + 1)) / ((2 * SQRT (2 + SQRT (3))) ^ 2-1 ^ 2) #

Позволять #sqrt (2 + SQRT (3)) = а #

# Rarrcot37.5 = (2 * (2 * а + 1) + SQRT (3) * (2 * а + 1)) / ((4 * (2 + SQRT (3))) ^ 2-1) #

# Rarrcot37.5 = (4а + 2 + 2sqrt (3) а + SQRT (3)) / ((4 * (2 + SQRT (3)) - 1) #

# Rarrcot37.5 = (4а + 2sqrt (3) а + а ^ 2) / (7 + 4sqrt (3)) * (7-4sqrt (3)) / (7-4sqrt (3)) #

# Rarrcot37.5 = 7 * (4a + 2sqrt (3) а + а ^ 2) -4sqrt (3) * (4a + 2sqrt (3) а + а ^ 2) #

# Rarrcot37.5 = 28а + 14sqrt (3) а + 7а ^ 2-16sqrt (3)-24а-4sqrt (3) ^ 2 #

# Rarrcot37.5 = 7а ^ 2-4sqrt (3) ^ 2 + 4а-2sqrt (3) #

# Rarrcot37.5 = а ^ 2 (7-4sqrt (3)) + 2 * а (2-SQRT (3)) #

# Rarrcot37.5 = (2 + SQRT (3)) (7-4sqrt (3)) + 2 * SQRT (2 + SQRT (3)) * SQRT ((2-SQRT (3))) * SQRT ((2 -sqrt (3))) #

# Rarrcot37.5 = 2 * (7-4sqrt (3)) + SQRT (3) (7-4sqrt (3)) + SQRT (2 ^ 2 * (2-SQRT (3))) #

# Rarrcot37.5 = 14-8sqrt (3) + 7sqrt (3) -12 + SQRT ((SQRT (6) -sqrt (2)) ^ 2) #

# Rarrtan52.5 = 2-SQRT (3) + SQRT (6) -sqrt (2) #

Доказанные.

Ответ:

Меньший подход …

Используемые правила: -

#color (красный) (ul (полоса (| цвет (зеленый) (sin2theta = 2 cdot sintheta cdot costheta)) | #

# Cos2theta = 2cos ^ 2theta-1 #

# => Цвета (красный) (ули (бар (| цвет (синий) (2cos ^ 2theta = 1 + cos2theta)) | #

Объяснение:

#tan (52,5 ^ @) #

# = Sin (52,5 ^ @) / соз (52,5 ^ @) #

# = Sin (105/2) ^ @ / соз (105/2) ^ @ #

# = (2 cdot sin (105/2) ^ @ cdot cos (105/2) ^ @) / (2 cdot cos (105/2) ^ @ cdot cos (105/2) ^ @ #

# = грех (105/2 xx2) ^ @ / (2 cdot cos ^ 2 (105/2) ^ @ #

# = Sin (105) ^ @ / (соз (105) ^ @ + 1) #

# = Sin (60 ^ @ + 45 ^ @) / (соз (60 ^ @ + 45 ^ @) + 1) #

# = (sin60 ^ @ cdot cos45 ^ @ + cos60 ^ @ cdot sin45 ^ @) / (cos60 ^ @ cdot cos45 ^ @ -sin60 ^ @ cdot sin45 ^ @ + 1 #

# = (sqrt3 / 2 cdot 1 / sqrt2 + 1/2 cdot 1 / sqrt2) / (1/2 cdot 1 / sqrt2-sqrt3 / 2 cdot 1 / sqrt2 + 1 #

# = ((Sqrt3 + 1) / (2sqrt2)) / ((1-sqrt3 + 2sqrt2) / (2sqrt2) #

# = (Sqrt3 + 1) / (1-sqrt3 + 2sqrt2 #

# = ((sqrt3 + 1) cdot (1 + 2sqrt2 + sqrt3)) / ((1 + 2sqrt2) ^ 2- (sqrt3) ^ 2) #

# = (Sqrt3 + 2sqrt6 + 3 + 1 + 2sqrt2 + sqrt3) / (1 + 4sqrt2 + 8-3) #

# = (2 (sqrt6 + sqrt3 + sqrt2 + 2)) / (6 + 4sqrt2) #

# = ((Sqrt6 + sqrt3 + sqrt2 + 2)) / (3 + 2sqrt2) #

# = ((3-2sqrt2) (sqrt6 + sqrt3 + sqrt2 + 2)) / ((3 + 2sqrt2) (3-2sqrt2) #

# = (Sqrt6-sqrt3-sqrt2 + 2) / 1 #

# = Sqrt6-sqrt3-sqrt2 + 2 #

Надеюсь, поможет…

Спасибо…

# Tan105 ^ @ = тангенс (60 ^ @ + 45 ^ @) #

# => Tan105 ^ @ = (tan60 ^ @ + tan45 ^ @) / # (1-tan60 ^ @ tan45 ^ @)

# => Tan105 ^ @ = (sqrt3 + 1) / (1-sqrt3 * 1) #

# => Tan105 ^ @ = (1 + sqrt3) / (1-sqrt3) #

# => Tan105 ^ @ = - ((sqrt3 + 1) (sqrt3-1)) / (sqrt3-1) ^ 2 #

# => Tan105 ^ @ = - (3-1) / (4-2sqrt3) #

# => (2tan52,5 ^ @) / (1-tan ^ 2 52,5 ^ @) = - 1 / (2-sqrt3) #

Позволять #tan52.5^@=x#

Сейчас

# (2x) / (1-х ^ 2) = - 1 / (2-sqrt3) #

# => Х ^ 2-2 (2-sqrt3) х-1 = 0 #

# => Х = (2 (2-sqrt3) + SQRT (4 (2-sqrt3) ^ 2 + 4)) / 2 #

Как #52.5^@in "Первый квадрант -вый корень забыт" #

# => Х = (2 (2-sqrt3) + 2sqrt ((2-sqrt3) ^ 2 + 1)) / 2 #

# => Х = (2-sqrt3) + SQRT ((2-sqrt3) ^ 2 + 1) #

# => Х = (2-sqrt3) + SQRT (8-4sqrt3) #

# => Х = (2-sqrt3) + SQRT (2 (4-2sqrt3) #

# => Х = (2-sqrt3) + SQRT (2 (sqrt3-1) ^ 2) #

# => Х = 2-sqrt3 + sqrt2 (sqrt3-1) #

# => Х = 2-sqrt3 + sqrt6-sqrt2 #

# => Х = sqrt6-sqrt3-sqrt2 + 2 #