Question

Is this an identity? 1+sec^2x=tan^2x

Answer 1

No, the correct identity is $\tan^2(x)+1=\sec^2(x)$. It is obtained by dividing the both sides of the Pythagorean Trigonometric Identity by cos²(x):

$\sin^2(x)+\cos^2(x)=1\\\\ \dfrac{\sin^2(x)+\cos^2(x)}{\cos^2(x)}=\dfrac{1}{\cos^2(x)}\\\\ \dfrac{\sin^2(x)}{\cos^2(x)}+\dfrac{\cos^2(x)}{\cos^2(x)}=\dfrac{1}{\cos^2(x)}\\\\ \boxed{\tan^2(x)+1=\sec^2(x)}~~\blacksquare$