How can the following identity be proved : (tan x cot x)^2 = sec^2 x + csc^2 x

We have to prove that (tan x * cot x)^2 = (sec x)^2 + (csc
x)^2.

Let's start with the righ hand
side:

(sec x)^2 + (csc
x)^2

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

=> [(sin x)^2 + (cos x)^2]/(sin x)^2 * (cos
x)^2

=> 1/(sin x)^2 * (cos
x)^2

The left hand side is (tan x * cot
x)^2

=> [(sin x)^2 / (cos x)^2]*[(cos x)^2 / (sin
x)^2]^2

=> 1

It is not
possible to equate (sin x)^2 * (cos x)^2 to
1.

So the given identity cannot be
proved.