The smallest positive value of x (in degrees) for which`tan(x+100^@)=tan(x+50^@).tanx.tan(x-50^@)` is

The smallest positive value of x (in degrees) for which`tan(x+100^@)=t

1.	The smallest positive value of x (in degrees) for which`tan(x+100^@)=tan(x+50^@).tanx.tan(x-50^@)` is
Answer» Correct Answer - `x = 30^(@)` `tan(x+100^(@))=tan(x+50^(@))tanxtan(x-50^(@))` `implies(tan(x+100^(@)))/(tanx)=tan(x+50^(@))tanxtan(x-50^(@)).` `implies(sin(x+100^(@)))/(cos(x+100^(@))).(cosx)/(sinx)=(sin(x+50^(@))sin(x-50^(@)))/(cos(x+50^(@))cos(x-50^(@)))` `implies(sin(2x+100^(@))+sin100^(@))/(sin(2x+100^(@))-sin100^(@))=(cos100^(@)-cos2x)/(cos100^(@)+cos2x)` `implies[sin(2x+100^(@))+sin100^(@)][cos100^(@)+cos2x]` `=[cos100^(@)-cos2x]xx[sin(2x+100^(@))-sin100^(@)]` `impliessin(2x+100^(@)).cos100^(@)+sin(2x+100^(@)).cos2x+sin100^(@)cos100^(@)+sin100^(@)cos2x` `=cos100^(@)sin(2x+100^(@))-cos100^(@)sin100^(@)-cos2xsin(2x+100^(@))+cos2xsin100^(@)` `implies2sin(2x+100^(@))cos2x+2sin100^(@)cos100^(@)=0` `impliessin(4x+100^(@))+sin100^(@)+sin200^(@)=0` `=sin(4x+100^(@))+2sin150^(@)cos50^(@)=0` `=sin(4x+100^(@))+2.(1)/(2)sin(90^(@)-50^(@))=0` `impliessin(4x+100^(@))+sin40^(@)=0` `impliessin(4x+100^(@))=sin(-40^(@))` `implies4x+100^(@)=npi+(-1)^(n)(-40^(@))` `implies4x=n(180^(@))+(-1)^(n)(-40^(@))-100^(@)` `impliesx=(1)/(4)[n(180^(@))+(-1)^(n)(-40^(@))-100^(@)]` The smallest positive value of x is obtained when n=1. Therefore, `x=(1)/(4)(180^(@)+40^(@)-100^(@))` `impliesx=(1)/(4)(120^(@))=30^(@)`