Question

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

Answer 1

`(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`

Answer 2

`+-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`