Identity: 1 + tan² = sec²

This identity links two trig ratios for the same angle: the tangent and the secant. It comes from the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\); if you divide every term by \(\cos^2\theta\), you get \(\tan^2\theta + 1 = \sec^2\theta\). The formula is true for every angle where \(\cos\theta \neq 0\), since both \(\tan\theta\) and \(\sec\theta\) must be defined.

Use this identity when you need to simplify trig expressions, prove another identity, or rewrite everything in terms of tangent or secant. It is also useful in calculus and physics when expressions contain \(\tan\theta\) and \(\sec\theta\) together.

• Forgetting that the identity is \(1 + \tan^2\theta = \sec^2\theta\), not \(1 + \tan\theta = \sec\theta\)
• Using the identity when \(\cos\theta = 0\), where tangent and secant are undefined
• Taking \(\sec\theta = \sqrt{1+\tan^2\theta}\) without considering that square roots can be positive or negative depending on the situation