입증 할 수있는 침대 (A / 2) - 3cot ((3A) / 2) = (4sinA) / (1 + 2cosA)?

대답:

자세한 내용은 설명.

설명:

우리는, # tan3theta = (3tantheta-tan ^ 3theta) / (1-3tan ^ 2theta) #.

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

(A / 2)} / {3tan (A / 2) -tan ^ 3 (A / 2)} #.

시키는 #tan (A / 2) = t, # 우리는,

#cot (A / 2) -3cot ((3A) / 2) #, # = 1 / t-3 {(1-3t ^ 2) / (3t-t ^ 3)} #, 1 / t- {3 (1-3t ^ 2)} / {t (3-t ^ 2)} #, # = {(3-t ^ 2) -3 (1-3t ^ 2)} / {t (3-t ^ 2)} #, # = (8t ^ 취소 (2)) / {취소 (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) 2)} #.

또한, (1 / tan ^ 2 (A / 2)) = sinA이며, # 2tan (A / 2)

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

#rArrcot (A / 2) -3cot ((3A) / 2) = (4sinA) / (1 + 2cosA), "원하는대로!"#

대답:

아래를 봐주세요.

설명:

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

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

sin (x / 2) * sin ((3x) / 2) * sin (x / 2)) / 2) #

2 * sin (x / 2) * sin ((3x) / 2) * sin (x / 2)) / 2) #

sin ((3x) / 2 + x / 2) - sin ((3x) / 2-x / 2) / (2x / 2)}) / (cos (3x) / 2-x / 2) -cos ((3x) / 2 + x / 2) #

sin ((4x) / 2) -sin ((2x) / 2)} / (cos ((2x) / 2) / 2) -cos ((4x) / 2) #

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

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

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

# = (4sinx (1-cosx)) / (cosx (1-cosx) + (1-cosx) (1 + cosx)

# = (4sinx (1- cosx)) / ((1- cosx) (cosx + 1 + cosx) #)

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

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

(3A) / 2) / sin ((3A) / 2) -2cot (A / 2) - cos (A / 2)

sin (A / 2) * sin ((3A) / 2) * sin (A / 2) 2)) - 2cot ((3A) / 2) #

(A) / (sin (A) / 2) # sin (A)

(3A / 2) # (2sin (A / 2) cos (A / 2)) / (sin (A / 2) * sin

2 × cos ((3A) / 2) / sin ((3A) / 2) # = 2cos (A / 2) / sin

2) cos (A / 2) -cos ((3A) / 2)) / sin ((3A) / 2)

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

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

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

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