Question

graph the expression 1-tan^2x/1+tan^2x, and make a conjecture about another expression that is equivalent to another.

Answer 1

Let us graph the function $y=\frac{1-\tan^2x}{1+\tan^2x}:$

It follows from the graph that $y=\cos(2x).$

Indeed, $\frac{1-\tan^2x}{1+\tan^2x} =\frac{1-\tan^2x}{\frac{1}{\cos^2 x}} =\cos^2x(1-\frac{\sin^2x}{\cos^2x}) =\cos^2x-\sin^2 x=\cos(2x).$