Simplification of [tex]\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}[/tex] is tan 4x
Solution:
By using sum-difference formula for tangent, Tangent of difference between two angles is written as,
[tex]\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}(\text { Equation } 1)[/tex]By comparing the above equation 1 with [tex]\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}[/tex] , we get x = 9x and y = 5x
By substituting x = 9x and y = 5x in equation 1,
[tex]\begin{array}{l}{\tan (9 x-5 x)=\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}} \\\\ {\tan (4 x)=\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}} \\\\ {\text { Therefore, } \frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}=\tan 4 x}\end{array}[/tex]Hence by using difference formula for tangent, [tex]\frac{\tan 9 x-\tan 5 x}{1+\tan 9 x \tan 5 x}[/tex] is simplified as tan4x