# LHS = cosec (x / 4) + cosec (x / 2) + cosecx #
# = cosec (x / 4) + cosec (x / 2) + cosecx + cotx-cotx #
# = cosec (x / 4) + cosec (x / 2) + color (azul) 1 / sinx + cosx / sinx -cotx #
# = cosec (x / 4) + cosec (x / 2) + color (azul) (1 + cosx) / sinx -cotx #
# = cosec (x / 4) + cosec (x / 2) + color (azul) (2cos ^ 2 (x / 2)) / (2sin (x / 2) cos (x / 2)) - cotx #
# = cosec (x / 4) + cosec (x / 2) + color (azul) (cos (x / 2) / sin (x / 2)) - cotx #
# = cosec (x / 4) + color (verde) (cosec (x / 2) + cuna (x / 2)) - cotx #
#color (magenta) "Procediendo de la misma manera que antes" #
# = cosec (x / 4) + color (verde) cuna (x / 4) -cotx #
# = cuna (x / 8) -cotx = RHS #
Responder:
Amablemente pasar por un Prueba dado en el Explicación.
Explicación:
Ajuste # x = 8y #, tenemos para probar que
# cosec2y + cosec4y + cosec8y = coty-cot8y #.
Observa eso, # cosec8y + cot8y = 1 / (sin8y) + (cos8y) / (sin8y) #, # = (1 + cos8y) / (sin8y) #, # = (2cos ^ 2 4y) / (2sin4ycos4y) #, # = (cos4y) / (sin4y) #.
# "Por lo tanto," cosec8y + co8y = cot4y = cot (1/2 * 8y) …….. (estrella) #.
Añadiendo, # cosec4y #, # cosec4y + (cosec8y + co8y) = cosec4y + cot4y #,
# = cuna (1/2 * 4y) ……… porque, (estrella) #.
#:. cosec4y + cosec8y + co8y = cot2y #.
Re-agregando # cosec2y # y reutilizando #(estrella)#, # cosec2y + (cosec4y + cosec8y + co8y) = cosec2y + cot2y #, # = cuna (1/2 * 2y) #.
#:. cosec2y + cosec4y + cosec8y + co8y = coty, es decir, #
# cosec2y + cosec4y + cosec8y = coty-cot8y #, ¡como se desee!
Responder:
Otro enfoque que parece haber aprendido previamente de respetado señor dk_ch.
Explicación:
# RHS = cuna (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)) = cancelar (sinx) / (cancelar (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 #
