Proving tan²x + 1 = sec²x

Trigonometry · Axiom Academy

EXAMPLE Proving tan²x + 1 = sec²x Derive a fundamental trigonometric identity step-by-step from the Pythagorean identity. Excellent work! You've successfully proven the identity tan²x + 1 = sec²x. Here's what we learned: Start with what you know: Many trigonometric identities can be derived from the fundamental Pythagorean identity sin²x + cos²x = 1 Strategic division: Dividing by cos²x was the key algebraic move that transformed the identity into one involving tangent and secant Recognize patterns: sin²x/cos²x = tan²x and 1/cos²x = sec²x are essential relationships to remember The identity family: This is one of three Pythagorean identities. Can you derive the other: 1 + cot²x = csc²x? Practical uses: This identity is crucial for simplifying trigonometric expressions and solving equations This proof technique—starting from a fundamental identity and using algebraic manipulation—is a powerful tool you'll use throughout trigonometry. Practice deriving other identities using similar strategies!

This is the written version of the interactive lesson above. See the full Trigonometry course.