Patunayan na ang cosec (x / 4) + cosec (x / 2) + cosecx = cot (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) + kulay (asul) 1 / sinx + cosx / sinx -cotx #

# = cosec (x / 4) + cosec (x / 2) + kulay (asul) (1 + cosx) / sinx -cotx #

# (cosc (x / 4) + cosec (x / 2) + kulay (asul) (2cos ^ 2 (x / 2)) / (2sin (x /

(x / 2) + kasalanan (x / 2)) - cotx #

# = cosec (x / 4) + kulay (green) (cosec (x / 2) + cot (x / 2)) - cotx #

#color (magenta) "Pagpapatuloy sa katulad na paraan tulad ng dati" #

# = cosec (x / 4) + kulay (green) cot (x / 4) -cotx #

# = cot (x / 8) -cotx = RHS #

Sagot:

Mahusay na dumaan sa isang Katunayan ibinigay sa Paliwanag.

Paliwanag:

Pagtatakda # x = 8y #, meron kami upang patunayan na,

# cosec2y + cosec4y + cosec8y = coty-cot8y #.

Sundin iyon, # cosec8y + cot8y = 1 / (sin8y) + (cos8y) / (sin8y) #, # = (1 + cos8y) / (sin8y) #, # = (2cos ^ 2 4y) / (2sin4ycos4y) #, # = (cos4y) / (sin4y) #.

# "Kaya," cosec8y + co8y = cot4y = cot (1/2 * 8y) …….. (bituin) #.

Pagdaragdag, # cosec4y #, # cosec4y + (cosec8y + co8y) = cosec4y + cot4y #,

# = cot (1/2 * 4y) ……… dahil, (bituin) #.

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

Muling pagdaragdag # cosec2y # at muling paggamit # (bituin) #, # cosec2y + (cosec4y + cosec8y + co8y) = cosec2y + cot2y #, # = cot (1/2 * 2y) #.

#:. cosec2y + cosec4y + cosec8y + co8y = coty, i.e., #

# cosec2y + cosec4y + cosec8y = coty-cot8y #, gaya ng ninanais!

Sagot:

Isa pang diskarte na tila ko na natutunan dati mula sa iginagalang sir dk_ch.

Paliwanag:

# RHS = cot (x / 8) -cotx #

# = cos (x / 8) / kasalanan (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 * kasalanan (x / 8) * cos (x / 8) * sinx) #

(x / 4)) / (sinx * sin (x / 4)) = kanselahin (sinx) / (kanselahin (sinx) * kasalanan (x / 4)) + (2sin ((3x (X / 4)) / (sinx * 2 * sin (x / 4) * cos (x / 4)) #

(x / 2)) / (sinx * sin (x / 2)) = cosecx + cosec (x / 2) + coesc (x / 4) = LHS #