להוכיח כי מיטה (A / 2) - 3cot ((3A) / 2) = (4sinA) / (1 + 2cosA)?

תשובה:

נא עיין הסבר.

הסבר:

אנחנו יודעים את זה, # tan3theta = (3tantheta-tan ^ 3theta) / (1-3tan ^ 2theta) # #.

#:. cot3theta = 1 / (tan3theta) = (1-3tan ^ 2theta) / (3tantheta-tan ^ 3theta) #

#:. cot (3A) / 2) = {1-3tan ^ 2 (A / 2)} / 3tan (A / 2) -tan ^ 3 (A / 2)} #.

לתת #tan (A / 2) = t, # יש לנו,

#cot (A / 2) -3 cot ((3A) / 2) #, # = 1 / t-3 {(1-3t ^ 2) / (3t-t ^ 3)} #, # 1 / t- {3 (1-3t ^ 2)} / {t (3-t ^ 2)} #, # (T-2 ^) -3 (1-3t ^ 2)} / {t (3-t ^ 2)} #, # = (8t ^ ביטול (2)) / {ביטול (t) (3-t ^ 2)} #, # = (8t) / {(1 + t ^ 2) +2 (1-t ^ 2)} #

(1 + t ^ 2) / (1 + t ^ 2) + 2 * (1-t ^ 2) / (1 + t ^ 2)} #.

שים לב ש, # (2t) / (1 + t ^ 2) = {2tan (A / 2)} / (1 + tan ^ 2 (A / 2)) = sinA, ו- #

# (1-t ^ 2) / (1 + t ^ 2) = cosa #.

# #AArccot / (1 + 2cosA), "לפי הצורך" # #

תשובה:

אנא ראה להלן.

הסבר:

# LHS = cot (x / 2) -3 cot ((3x) / 2) #

# 3 cos (3x) / 2 / חטא (3x) / 2) # # cos (x / 2) / sin (x / 2) -3 *

# (3x) / 2) * (c / x / 2) * חטא (3x) /) / 2) #

# (2xin) (3x) / 2) * cos (x / 2) -3 * 2cos (3x) / 2) * חטא (x / 2)) / (2sin (x / 2) * חטא (3x) / 2) #

# 3 (חטא) (3x) / 2 + x / 2) + חטא (3x) / 2-x / 2) -3 * {חטא (3x) / 2 + x / 2) / 2-x / 2)} / cos (3x) / 2-x / 2) -cos ((3x) / 2 + x / 2) #

(2x) / 2)} (cx (2x) (= 4x) / 2) + חטא (2x) / 2) -3 * / 2) - cos ((4x) / 2) #

# = (sin2x + sinx-3sin2x + 3sinx) / (cosx-cos2x) #

# = (4xinx-2sin2x) / (cosx- (cos ^ 2x-sin ^ 2x) # #

# = (4xinx-4sinx * cosx) / (cosx-cos ^ 2x + sin = 2x) #

# (4xinx (1-cosx)) / (cosx (1-cosx) + (1-cosx) (1 + cosx)) #

# (4xinx (1-cosx)) / (1-cosx) (cosx + 1 + cosx) #

# = (4sinx) / (1 + 2cosx) = RHS #

# LHS = cot (A / 2) -3 cot ((3A) / 2) #

# cos (A / 2) / חטא (A / 2) -Cos ((3A) / 2) / חטא (3A) / 2) -2 cot (3A) / 2) #

# (חטא) (3 א) / 2) * cos (A / 2) - cos (3A) / 2) * חטא (A / 2)) / (חטא (A / 2) * חטא (3A) / 2)) - 2cot ((3A) / 2) #

# (חטא = A) / חטא (A / 2) * חטא (3A) / 2)) -2 (3A) / 2)

# (2 / A) 2 (C / A) 2 / (c / A) 2 / (חטא (A / 2 * חטא (3A) / 2)) -2 (3A) / 2)

# 3 cos (A / 2) / חטא (3A) / 2) -2 * cos (3A) / 2) / חטא (3 א) / 2) # #

# (2) (cos (A / 2-cos ((3A) / 2)) / חטא ((3 A / / 2) #

# # 2 (2sin (A / 2) A / 2) / (3sin (A / 2) -4 sin ^ 3 (A / 2)

# (4sin (A / 2) חטא (א)) / (חטא (A / 2) (3-4sin ^ 2 (A / 2)) #

# = (4sin (A)) / (3-2 (1-cosA)) #

# = (4sin (A)) / (1 + 2cosA) = RHS #