Buktikan bahawa 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) + warna (biru) 1 / sinx + cosx / sinx -cotx #

# = cosec (x / 4) + cosec (x / 2) + warna (biru) (1 + cosx) / sinx -cotx #

(x / 2)) / (2sin (x / 2) cos (x / 2)) - cotx #

# = cosec (x / 4) + cosec (x / 2) + warna (biru) (cos (x / 2) / sin (x /

# = cosec (x / 4) + warna (hijau) (cosec (x / 2) + cot (x / 2)) - cotx #

#color (magenta) "Teruskan dengan cara yang sama seperti dahulu" #

# = cosec (x / 4) + warna (hijau) katil (x / 4) -cotx #

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

Jawapan:

Sila pergi melalui a Bukti diberikan dalam Penjelasan.

Penjelasan:

Menetapkan # x = 8y #, kita ada untuk membuktikan bahawa,

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

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

# "Oleh itu," cosec8y + co8y = cot4y = cot (1/2 * 8y) …….. (bintang) #.

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

# = cot (1/2 * 4y) ……… kerana, (bintang) #.

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

Tambah semula # cosec2y # dan menggunakan semula # (bintang) #, # cosec2y + (cosec4y + cosec8y + co8y) = cosec2y + cot2y #, # = cot (1/2 * 2y) #.

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

# cosec2y + cosec4y + cosec8y = coty-cot8y #, seperti yang dikehendaki!

Jawapan:

Pendekatan lain saya seolah-olah telah belajar dari sebelumnya dihormati tuan dk_ch.

Penjelasan:

# RHS = cot (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 /

# = (2sin ((7x) / 8) * cos (x / 8)) / (2 * sin (x / 8) * cos (x / 8) * sinx)

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

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