# LHS = cosec (x / 4) + cosec (x / 2) + cosecx #
# = cosec (x / 4) + cosec (x / 2) + cosecx + cotx-cotx #
# = cosec (x / 4) + cosec (x / 2) + barva (modra) 1 / sinx + cosx / sinx -cotx #
# = cosec (x / 4) + cosec (x / 2) + barva (modra) (1 + cosx) / sinx -cotx #
# = cosec (x / 4) + cosec (x / 2) + barva (modra) (2cos ^ 2 (x / 2)) / (2sin (x / 2) cos (x / 2)) - cotx #
# = cosec (x / 4) + cosec (x / 2) + barva (modra) (cos (x / 2) / sin (x / 2)) - cotx #
# = cosec (x / 4) + barva (zelena) (cosec (x / 2) + posteljica (x / 2)) - cotx #
#color (magenta) "Nadaljevanje na podoben način kot prej" #
# = cosec (x / 4) + barvna (zelena) posteljica (x / 4) -cotx #
# = posteljica (x / 8) -cotx = RHS #
Odgovor:
Vljudno gremo skozi Dokaz v. t Razlaga.
Pojasnilo:
Nastavitev # x = 8y #, imamo to dokazati,
# cosec2y + cosec4y + cosec8y = coty-cot8y #.
Opazujte, da # cosec8y + cot8y = 1 / (sin8y) + (cos8y) / (sin8y) #, # = (1 + cos8y) / (sin8y) #, # = (2cos ^ 2 4y) / (2sin4ycos4y) #, # = (cos4y) / (sin4y) #.
# "Torej," cosec8y + co8y = cot4y = posteljica (1/2 * 8y) …….. (zvezda) #.
Dodajanje, # cosec4y #, # cosec4y + (cosec8y + co8y) = cosec4y + cot4y #,
# = otroška posteljica (1/2 * 4y) ……… ker, (zvezda) #.
#:. cosec4y + cosec8y + co8y = cot2y #.
Ponovno dodajanje # cosec2y # in ponovno uporabo #(zvezda)#, # cosec2y + (cosec4y + cosec8y + co8y) = cosec2y + cot2y #, # = posteljica (1/2 * 2y) #.
#:. cosec2y + cosec4y + cosec8y + co8y = coty, tj. #
# cosec2y + cosec4y + cosec8y = coty-cot8y #, po želji!
Odgovor:
Zdi se, da sem se že prej naučil drugega pristopa spoštovan gospod dk_ch.
Pojasnilo:
# RHS = posteljica (x / 8) -cotx #
# = cos (x / 8) / sin (x / 8) -cosx / sinx #
# = (sinx * cos (x / 8) -cosx * sin (x / 8)) / (sinx * sin (x / 8)) #
# = sin (x-x / 8) / (sinx * sin (x / 8)) = sin ((7x) / 8) / (sinx * sin (x / 8)) #
# = (2sin ((7x) / 8) * cos (x / 8)) / (2 * sin (x / 8) * cos (x / 8) * sinx) #
# = (sinx + sin ((3x) / 4)) / (sinx * sin (x / 4)) = preklic (sinx) / (odpoved (sinx) * sin (x / 4)) + (2sin ((3x) / 4) * cos (x / 4)) / (sinx * 2 * sin (x / 4) * cos (x / 4)) #
# = cosec (x / 4) + (sinx + sin (x / 2)) / (sinx * sin (x / 2)) = cosecx + cosec (x / 2) + coesc (x / 4) = LHS #
