Jawapan:
# sin ^ 2theta-csc ^ 2theta = -8sqrt3 #
Penjelasan:
Di sini, Jika # sinθ + cosecθ = 4 #, kemudian # sin ^ 2θ-cosec ^ 2θ =? #
Biarkan
#color (biru) (sintheta + csctheta = 4 … ke (1) #
Squaring both sides
# (sintheta + csctheta) ^ 2 = 4 ^ 2 #
# => sin ^ 2theta + 2sinthetacsctheta + csc ^ 2theta = 16 #
# => sin ^ 2theta + csc ^ 2theta = 16-2sinthetacsctheta #
Menambah,#color (hijau) (- 2sinthetacsctheta # kedua-dua pihak
# sin ^ 2theta-2sinthetacsctheta + csc ^ 2theta = 16- 4sinthetacsctheta #
# (sintheta-csctheta) ^ 2 = 16-4, di mana, warna (hijau) (sinthetacsctheta = 1 #
# (sintheta-csctheta) ^ 2 = 12 = (4xx3) = (2sqrt3) ^ 2 #
# sintheta-csctheta = + - 2sqrt3 #
Tetapi, #color (merah) (- 1 <= sintheta <= 1 dan sintheta + csctheta = 4 #
#: warna (merah) (1 <= csctheta <= 4 => sintheta <csctheta => sintheta-csctheta <0 #
Jadi, #color (biru) (sintheta-csctheta = -2sqrt3 … ke (2) #
Dari #color (biru) ((1) dan (2) #,kita mendapatkan
# sin ^ 2theta-csc ^ 2theta = (sintheta + csctheta) (sintheta-csctheta) #
# sin ^ 2theta-csc ^ 2theta = (4) (- 2sqrt3) #
# sin ^ 2theta-csc ^ 2theta = -8sqrt3 #
