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