Auxiliary trigonometric identities [Identidades Trigonométricas Auxiliares] I need to show two auxiliary trigonometric identities:
1) $\sec^2x = \tan ^2x + 1 (\cos x \neq 0)$
2) $\csc^2x = \cot^2x +1 (\sin x \neq 0)$
How could I do it?

[Original Portuguese]
Identidades Trigonométricas Auxiliares
Sendo preciso mostrar a demonstração das duas fórmulas das IDENTIDADES TRIGONOMÉTRICAS AUXILIARES:
1) sec²x = tg²x + 1 (cos x ≠ 0)
2) cossec²x = cotg²x = 1 (sen x ≠ 0)
Como seria possível fazê-lo???

Starting from $\sin^2 x + \cos^2 x = 1$, you get, dividing by $\cos^2 x$ (if this is $\ne 0$)
\[ \frac{\sin^2x}{\cos^2 x} + \frac{\cos^2 x}{\cos^2 x} = \frac 1{\cos^2 x} \]
which gives
\[  \tan^2 x + 1 = \mathrm{sec}^2 x \]
Dividing the first identity by $\sin^2 x$ we get
\[ \frac{\sin^2x}{\sin^2 x} + \frac{\cos^2 x}{\sin^2 x} = \frac 1{\sin^2 x} \]
which gives
\[  1 + \cot^2 x = \mathrm{csc}^2 x \]
