Sagot:
# sin ^ 2theta-csc ^ 2theta = -8sqrt3 #
Paliwanag:
Dito, Kung # sinθ + cosecθ = 4 #, pagkatapos # sin ^ 2θ-cosec ^ 2θ =? #
Hayaan
#color (blue) (sintheta + csctheta = 4 … to (1) #
Squaring both sides
# (sintheta + csctheta) ^ 2 = 4 ^ 2 #
# => sin ^ 2theta + 2sinthetacsctheta + csc ^ 2theta = 16 #
# => sin ^ 2theta + csc ^ 2theta = 16-2sinthetacsctheta #
Pagdaragdag,#color (green) (- 2sinthetacsctheta # magkabilang panig
# sin ^ 2theta-2sinthetacsctheta + csc ^ 2theta = 16- 4sinthetacsctheta #
# (sintheta-csctheta) ^ 2 = 16-4, kung saan, kulay (berde) (sinthetacsctheta = 1 #
# (sintheta-csctheta) ^ 2 = 12 = (4xx3) = (2sqrt3) ^ 2 #
# sintheta-csctheta = + - 2sqrt3 #
Ngunit, #color (pula) (- 1 <= sintheta <= 1 at sintheta + csctheta = 4 #
#:. color (red) (1 <= csctheta <= 4 => sintheta <csctheta => sintheta-csctheta <0 #
Kaya, #color (asul) (sintheta-csctheta = -2sqrt3 … sa (2) #
Mula sa #color (asul) ((1) at (2) #, makuha namin
# sin ^ 2theta-csc ^ 2theta = (sintheta + csctheta) (sintheta-csctheta) #
# sin ^ 2theta-csc ^ 2theta = (4) (- 2sqrt3) #
# sin ^ 2theta-csc ^ 2theta = -8sqrt3 #
