Dokazati da cosec (x / 4) + cosec (x / 2) + cosecx = krevetić (x / 8) -cotx?

# LHS = cosec (x / 4) + cosec (x / 2) + cosecx #

# = cosec (x / 4) + cosec (x / 2) + cosecx + cotx-cotx #

# = cosec (x / 4) + cosec (x / 2) + boja (plava) 1 / sinx + cosx / sinx -cotx #

# = cosec (x / 4) + cosec (x / 2) + boja (plava) (1 + cosx) / sinx -cotx #

# = cosec (x / 4) + cosec (x / 2) + boja (plava) (2cos ^ 2 (x / 2)) / (2sin (x / 2) cos (x / 2)) - cotx #

# = cosec (x / 4) + cosec (x / 2) + boja (plava) (cos (x / 2) / sin (x / 2)) - cotx #

# = cosec (x / 4) + boja (zelena) (cosec (x / 2) + krevetić (x / 2)) - cotx #

#color (magenta) "Nastavlja se na sličan način kao prije" #

# = Cosec (x / 4) + boje (zeleno) krevetić (x / 4) -cotx #

# = Krevetić (x / 8) -cotx = RHS #

Odgovor:

Ljubazno prođite kroz Dokaz dane u Obrazloženje.

Obrazloženje:

postavljanje # x = 8y #, imamo to dokazati,

# Cosec2y + cosec4y + = cosec8y Coty-cot8y #.

Promatrajte to, # Cosec8y + cot8y = 1 / (sin8y) + (cos8y) / (sin8y) #, # = (1 + cos8y) / (sin8y) #, # = (2cos ^ 2 4y) / (2sin4ycos4y) #, # = (Cos4y) / (sin4y) #.

# "Dakle," cosec8y + co8y = cot4y = krevetić (1/2 * 8y) …….. (zvijezda) #.

Dodavanje, # Cosec4y #, # Cosec4y + (cosec8y + co8y) = + cosec4y cot4y #,

# = krevetić (1/2 * 4y) ……… jer, (zvijezda) #.

#:. cosec4y + cosec8y + = co8y cot2y #.

Ponovnog dodavanja # Cosec2y # i ponovno korištenje #(zvijezda)#, # Cosec2y + (cosec4y + cosec8y + co8y) = + cosec2y cot2y #, # = Dječji (1 / 2x 2y) #.

#:. cosec2y + cosec4y + cosec8y + co8y = coty, tj., #

# cosec2y + cosec4y + cosec8y = coty-cot8y #, po želji!

Odgovor:

Još jedan pristup koji sam ranije naučio poštovani sir dk_ch.

Obrazloženje:

# RHS = krevetić (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)) / (2x sin (x / 8) + cos (x / 8) * sinx) #

# = (Sinx + sin ((3 x) / 4)) / (sinx * sin (x / 4)) = poništavanje (sinx) / (otkazivanje (sinx) + sin (x / 4)) + (2sin ((3 x) / 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 #