Determine the smallest positive value of x ( in degrees ) for which `tan(x+100^(@))=tan (x+50^(@))tan x tan (x-50^( @))`

Explanation

`tan(x+100^(@))`
`=tan(x+50^(@))tanx. Tan(x-50^(@))`
`rArr sin(x+100^(@))cos(x-50^(@))/(cos(x+100^(@))sin(x-50^(@))`
`=sin(x+50^(@)sinx)/(cos(x+50^(@))cosx)`
`rArr 2sin(x+100^(@))cos(x-50^(@))/(2cos(x+50^(@))cosx)`
`rArr (sin(2x+50^(@))+sin150^(@))/(sin(2x+50^(@))-sin150^(@))`
`rArr =(cos50^(@)-cos(2x+50^(@)))/(cos(2x+50^(@))+cos50^(@))`
`rArr [sin(2x+50^(@))+sin150^(@)]`
`([cos50^(@)-cos(2x+50^(@)])=[sin(2x+50^(@))+cos50^(@))`
`rArr sin(2x+50^(@))cos(2x+50^(@))+cos50^(@)sin(2x+50^(@))+sin 150^(@)cos(2x+50^(@))+sin150^(@)cos50^(@)`
`=cos(2x+50^(@))-sin150^(@)cos(2x+50^(@))`
`rArr 2sin(2x+50^(@))cos(2x+50^(@))= -2sin180^(@)cos50^(@)`
`rArr sin(4x+100^(@))=-2 xx sin (180^(@)-30^(@))cos50^(@)`
`=-2(1/2)cos50^(@)`
`=-cos50^(@)`
`sin(270^(@)-50^(@))=sin220^(@)`
`rArr 4x+100^(@)=220^(@)`
`rArr x=30^(@)`. Ans.