Is tan²x ? sec²x - 1 an identity?

Yes, sec²x−1=tan²x is an identity.

sec²−1=tan²x

Let us derive the equation

We know the identity

sin²(x)+cos²(x)=1 ——-(i)

Dividing throughout the equation by cos²(x)

We get

sin²(x)/cos²(x) + cos²(x)/cos²(x) = 1/cos²(x)

We know that

sin²(x)/cos²(x)= tan²(x), and cos²(x)/cos²(x) = 1

So the equation (i) after substituting becomes

tan²(x) +1= 1/cos²(x) ——–(ii)

Now we know that 1/cos²(x)= sec²(x)

So on substitution equation (ii) becomes

tan²(x) +1= sec²(x)

On rearranging the terms we get

sec²x−1=tan²x

Hence Proved