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