sec^2x+tan^2xsec^2x=sec^4x

To prove that sec^2x + tan^2x * sec^2x = sec^4x, we need to simplify both sides of the equation using the basic trigonometric identities.

Let's start with the left-hand side (LHS) of the equation:

LHS: sec^2x + tan^2x * sec^2x

First, we know that tan^2x can be expressed in terms of sec^2x:

tan^2x = sec^2x - 1

We can substitute this into the original equation:

LHS: sec^2x + (sec^2x - 1) * sec^2x

Now, let's simplify the equation further:

LHS: sec^2x + (sec^4x - sec^2x)
LHS: sec^4x - sec^2x + sec^2x
LHS: sec^4x

As you can see, the left-hand side (LHS) simplifies to sec^4x, which is the same as the right-hand side (RHS) of the equation. Therefore, we have shown that sec^2x + tan^2x * sec^2x = sec^4x.

In summary, to prove this relationship, we made use of the identity tan^2x = sec^2x - 1 and then simplified the equation step by step until we reached the desired result.