Here,
If #sinθ+cosecθ=4#, then #sin^2θ-cosec^2θ =?#
Let
#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#
Adding ,#color(green)(-2sinthetacsctheta # both sides
#sin^2theta-2sinthetacsctheta+csc^2theta=16-
4sinthetacsctheta#
#(sintheta-csctheta)^2=16-4 ,where, color(green)(sinthetacsctheta=1#
#(sintheta-csctheta)^2=12=(4xx3)=(2sqrt3)^2#
#sintheta-csctheta=+-2sqrt3#
But, #color(red)(-1 <= sintheta <= 1 and sintheta+csctheta=4#
#:.color(red)(1 <= csctheta <= 4=>sintheta < csctheta=>sintheta-csctheta <
0#
So,
#color(blue)(sintheta-csctheta=-2sqrt3...to(2)#
From #color(blue)((1)and(2)#,we get
#sin^2theta-csc^2theta=(sintheta+csctheta)(sintheta-csctheta)#
#sin^2theta-csc^2theta=(4)(-2sqrt3)#
#sin^2theta-csc^2theta=-8sqrt3#
