Αποδείξτε ότι η κούνια (A / 2) - 3cot ((3A) / 2) = (4sinA) / (1 + 2cosA);

Απάντηση:

Ανατρέξτε στο Εξήγηση.

Εξήγηση:

Ξέρουμε ότι, # tan3theta = (3antheta-tan ^ 3theta) / (1-3tan ^ 2theta) #.

#:. (3tantheta-tan ^ 3theta) = (3tane ^ 2theta)

(3Α) / 2) = {1-3tan ^ 2 (Α / 2)} / {3tan (A / 2).

Εκμίσθωση # ttan (Α / 2) = t, # έχουμε,

#cot (Α / 2) -3cot ((3Α) / 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, = {(8t ^ ακυρώστε (2)) / {ακυρώστε (t) (3-t ^ 2), # = (8t) / {(1 + t ^ 2) + 2 (1-t ^ 2)

(1 + t2) / (1 + t ^ 2) / 2 * (1-t ^ 2) / (1 + 2)} #.

Σημειώστε ότι, # (2t) / (1 + t ^ 2) = {2tan (A / 2)} /

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

#rArrcot (Α / 2) -3cot ((3Α) / 2) = (4sinA) / (1 + 2cosA)

Απάντηση:

Παρακαλούμε δείτε παρακάτω.

Εξήγηση:

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

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

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

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

= (sin ((3χ) / 2 + χ / 2) -σιν ((3x) / 2 + / 2-x / 2)}) / (cos ((3χ) / 2-χ / 2)

= (sin ((4x) / 2)-sin ((2x) / 2) / 2) -cos ((4χ) / 2) #

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

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

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

= Cosx (1-cosx) + (1-cosx) (1 + cosx)) #

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

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

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

(3Α) / 2) / sin ((3Α) / 2) -2cot ((3Α) / 2) # = cos (A / 2)

= (sin ((3A) / 2) * cos (A / 2) -cos ((3A) / 2) * sin (A / 2) 2)) - 2cot ((3Α) / 2) #

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

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

= 2cos (A / 2) / sin ((3Α) / 2) -2 * cos ((3Α) / 2)

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

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

= (4sin (A / 2) sin (A)) / (sin (A / 2)

# = (4sin (Α)) / (3-2 (1-οοοΑ)) #

# = (4sin (Α)) / (1 + 2cosA) = RHS #