If sinθ+cosecθ=4 Then sin^2θ-cosec^2θ =?

1 Answer
Apr 27, 2018

sin^2theta-csc^2theta=-8sqrt3sin2θcsc2θ=83

Explanation:

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