Odgovor:
# sin ^ 2theta-csc ^ 2theta = -8sqrt3 #
Pojasnilo:
Tukaj, Če # sinθ + cosecθ = 4 #, potem # sin ^ 2θ-cosec ^ 2θ =? #
Let
#barva (modra) (sintheta + csctheta = 4 … do (1) #
Kvadriranje obeh strani
# (sintheta + csctheta) ^ 2 = 4 ^ 2 #
# => sin ^ 2theta + 2sinthetacsctheta + csc ^ 2theta = 16 #
# => sin ^ 2theta + csc ^ 2theta = 16-2sintetacsctheta #
Dodajanje,#color (zelena) (- 2sintetacsctheta # obe strani
# sin ^ 2theta-2sinthetacsctheta + csc ^ 2theta = 16-4sintetacsctheta #
# (sintheta-csctheta) ^ 2 = 16-4, kje, barva (zelena) (sinthetacsctheta = 1 #
# (sintheta-csctheta) ^ 2 = 12 = (4xx3) = (2sqrt3) ^ 2 #
# sintheta-csctheta = + - 2sqrt3 #
Ampak, #barva (rdeča) (- 1 <= sintheta <= 1 in sintheta + csctheta = 4 #
#:. barva (rdeča) (1 <= csctheta <= 4 => sintheta <csctheta => sintheta-csctheta <0 #
Torej, #color (modra) (sintheta-csctheta = -2sqrt3 … do (2) #
Od #barva (modra) ((1) in (2) #,dobimo
# sin ^ 2theta-csc ^ 2theta = (sintheta + csctheta) (sintheta-csctheta) #
# sin ^ 2theta-csc ^ 2theta = (4) (- 2sqrt3) #
# sin ^ 2theta-csc ^ 2theta = -8sqrt3 #
