I am right to say that the identity of sec^2 is 1/sin^2
The above statement is not an identity because the quantities you wrote on each side of the = are not the same.
$\displaystyle sec^{7}x = \frac{1}{cos^{2}x}$
which would be the same as writing down
$\displaystyle \left (\frac{1}{cos(x)} \right ) ^{7} = \left ( \frac{1}{cos(x)} \right )^{2}$
which is incorrect
If you have the proper identity
$\displaystyle sec(x) = \frac{1}{cos}$
both sides need to be raised to the same power to maintain the identity.