Verify the identity 1=(1+tan^2 x)*cos^2 x

We notice that inside brackets we have a consequence of
Pythagorean identity:

1 + (tan x)^2 = 1/(cos
x)^2

We'll show how it
works:

Pythagorean identity states
that:

(sin x)^2 + (cos x)^2 =
1

We'll divide by (cos x)^2:

1
+ (sin x)^2/ (cos x)^2 = 1/(cos x)^2

But (sin x)^2/ (cos
x)^2 = (tan x)^2

1 + (tan x)^2= 1/(cos
x)^2

Now, we'll substitute what's inside brackets by the
equivalent above:

(cos x)^2*[1 + (tan x)^2] =
(cos x)^2*(1/(cos x)^2) = 1