No, the correct identity is [tex]\tan^2(x)+1=\sec^2(x)[/tex]. It is obtained by dividing the both sides of the Pythagorean Trigonometric Identity by cos²(x):
[tex]\sin^2(x)+\cos^2(x)=1\\\\
\dfrac{\sin^2(x)+\cos^2(x)}{\cos^2(x)}=\dfrac{1}{\cos^2(x)}\\\\
\dfrac{\sin^2(x)}{\cos^2(x)}+\dfrac{\cos^2(x)}{\cos^2(x)}=\dfrac{1}{\cos^2(x)}\\\\
\boxed{\tan^2(x)+1=\sec^2(x)}~~\blacksquare[/tex]
