यदि `tan (x +100^(@)) = tan ( x +50^(@)) tan x * tan ( x- 50^(@))` तो x का न्यूनतम धनात्मक मान ज्ञात कीजिए।

Explanation

` tan ( x - 100^(@)) = tan (x + 50^(@)) tan x `
` tan ( x- 50^(@))`
` rArr (tan (x + 100^(@)))/(tan (x - 50^(@)) )= tan (x + 50^(@)) tan x `
` rArr ( sin (x+ 100^(@))cos(x-50^(@)))/(cos(x+100^(@))sin(x-50^(@)))`
` = (sin (x+ 50^(@))sin x)/(cos (x+50^(@))cosx)`
` rArr (2 sin ( x+ 100^(@))cos(x- 50^(@)))/(2 cos (x+ 100^(@)) sin (x- 50^(@)))`
` = (2 sin (x+50^(@))sinx)/(2cos (x+50^(@) ) cosx)`
` rArr (sin (2x+50^(@))+sin150^(@))/(sin(2x+50^(@))- sin 150^(@) )`
` = ( cos 50^(@) - cos ( 2x + 50^(@)))/(cos (2x +50^(@)) + cos 50^(@))`
`rArr [ sin ( 2x + 50^(@))+ sin 150^(@)] `
` [ cos ( 2x + 50^(@) ) + cos 50^(@)]`
` [ cos 50^(@) - cos (2x + 50^(@))]`
`rArr sin (2x+50^(@))cos(2x+50^(@))+cos50^(@)sin(2x+50^(@))+sin150^(@)cos(2x+50^(@))+sin150^(@)cos50^(@)`
`=sin(2x+50^(@))cos50^(@)-sin(2x+50^(@))`
`cos(2x+50^(@))-sin150^(@)cos50^(@)+sin150^(@)cos(2x+50^(@))`
`rArr" "2sin (2x+50^(@))cos(2x+50^(@))`
`" "=-2sin150^(@)cos50^(@)`
`rArr" "sin(4x+100^(@))=-2xx sin(180^(@)-30^(@))cos50^(@)`
`" "=-2sin30^(@)cos 50^(@)`
`=-2((1)/(2))cos 50^(@)`
`=-cos50^(@)`
`=sin (270^(@)-50^(@))=sin220^(@)`
`rArr" "4x+100^(@)=220^(@)`
`rArr" "x=30^(@)`.