7cosec theta -3 cot theta =77cosecθ3cotθ=7,then what is the value of 7cot theta-3 cosec theta cotθ3cosecθ ?

2 Answers
Nov 14, 2016

(7csctheta-3cottheta)^2-(7cottheta-3csctheta)^2(7cscθ3cotθ)2(7cotθ3cscθ)2

=49(csc^2theta-cot^2theta)-9(csc^2theta-cot^2theta)-42cscthetacottheta+42cscthetacottheta=49(csc2θcot2θ)9(csc2θcot2θ)42cscθcotθ+42cscθcotθ

=49-9=40=499=40

So
(7csctheta-3cottheta)^2-(7cottheta-3csctheta)^2=40(7cscθ3cotθ)2(7cotθ3cscθ)2=40

=>7^2-(7cottheta-3csctheta)^2=4072(7cotθ3cscθ)2=40

=>(7cottheta-3csctheta)^2=49-40=9(7cotθ3cscθ)2=4940=9

=>7cottheta-3csctheta=+-37cotθ3cscθ=±3

Nov 14, 2016

+-3±3

Explanation:

7/sintheta-3costheta/sintheta=77sinθ3cosθsinθ=7

7-3costheta = 7sintheta73cosθ=7sinθ

Put X=costhetaX=cosθ and Y=sinthetaY=sinθ then X^2+Y^2=1X2+Y2=1 and 7-3X=7Y73X=7Y

so X^2+(1-3/7X)^2=1X2+(137X)2=1

X^2+cancel(1)+9/49X^2-6/7X=cancel(1)

X=0 or 58/7X=6 => X=21/29

The solution X=0 gives theta=pi/2 so

7cot (pi/2)-3csc(pi/2)=0-3=-3

For theta=arccos(21/29) then costheta=21/29, sintheta=20/29

so

7costheta/sintheta-3/sintheta=((7*21)/29-3)/(20/29)=(147-87)/20=3