The given question is
[tex]\frac{4\tan x}{1+\tan ^2x}[/tex]
Use the identity
[tex]1+\tan ^2x=\sec ^2x[/tex]
Then replace the denominator by sec^2 (x)
[tex]\frac{4\tan x}{\sec ^2x}[/tex]
Since sec is the reciprocal of cos, then
[tex]\sec ^2x=\frac{1}{\cos ^2x}[/tex]
Replce sec^2(x) by 1/cos^2(x)
[tex]\frac{4\tan x}{\frac{1}{\cos ^2x}}[/tex]
Since denominator of denominator will be a numerator
[tex]4\tan x\times\cos ^2x[/tex]
Use the value of tan
[tex]\tan x=\frac{\sin x}{\cos x}[/tex]
Replace tan by sin/cos
[tex]4\times\frac{\sin x}{\cos x}\times\cos ^2x[/tex]
Reduce cos(x) up with cos(x) down
[tex]\begin{gathered} 4\times\sin x\times\cos x= \\ 4\sin x\cos x \end{gathered}[/tex]
Use the identity
[tex]\sin (2x)=2\sin x\cos x[/tex][tex]4\sin x\cos x=2(2\sin x\cos x)[/tex]
Replace 2 sin(x)cos(x) by sin(2x)
[tex]2(2\sin x\cos x)=2\sin 2x[/tex]
The answer is
2 sin(2x)
