Cot x sec4x = cot x + 2 tan x + tan3x

Is this equation or identity?

The correct answer is:

An equation.

Explanation:

An equation is a statement that is true for particular values of the variable. An identity, however, is a statement that is true for any values of the variable.

If we use 60° as x, the left hand side of the equation gives us - 1.15. The right hand side gives us 4.04. These are not equal, so this cannot be an identity.

It is an equation:

here is the proof:

The key to this one is that sec^2 (x) = 1 + tan^2 (x).

So the left side is

cot (x) (1 + tan^2 (x)) (1 + tan^2 (x))

Expanding this term by term you get

cot (x) + 2 cot (x) tan^2 (x) + cot (x) tan^4 (x),

but since cot (x) tan (x) = 1, that turns into

cot (x) + 2 tan (x) + tan^3 (x),

which is the same as the right side.