#7cosec theta -3 cot theta =7#,then what is the value of 7#cot theta-3 cosec theta # ?

2 Answers
Nov 14, 2016

#(7csctheta-3cottheta)^2-(7cottheta-3csctheta)^2#

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

#=49-9=40#

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

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

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

#=>7cottheta-3csctheta=+-3#

Nov 14, 2016

Answer:

#+-3#

Explanation:

#7/sintheta-3costheta/sintheta=7#

#7-3costheta = 7sintheta#

Put #X=costheta# and #Y=sintheta# then #X^2+Y^2=1# and #7-3X=7Y#

so #X^2+(1-3/7X)^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#