Question

verify the trig identity 1-cos^2=tan(x)cos(x)sin(x)=(-infinite, +infinite)

Answer 1

Notice that

$\tan x\cos x\sin x$

Is equal to

$\begin{gathered} \tan x\cos x\sin x=\frac{\sin x}{\cos x}\cos x\sin x=\sin x\sin x=\sin ^2x \\ \Rightarrow\tan x\cos x\sin x=\sin ^2x \end{gathered}$

And,

$\cos ^2x+\sin ^2x=1$

Thus,

$\begin{gathered} \Rightarrow\sin ^2x=1-\cos ^2x \\ \Rightarrow\tan x\cos x\sin x=1-\cos ^2x \end{gathered}$

However, notice that this identity requires that cosx is different than zero; otherwise, we would not be able to define tanx.