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

1 Answer
Apr 27, 2018

#sin^2theta-csc^2theta=-8sqrt3#

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#