Proving (1 - cos x)(1 + cos x) = sin²x

Trigonometry · Axiom Academy

EXAMPLE Proving Trigonometric Identities Learn to prove (1 - cos x)(1 + cos x) = sin²x using algebraic patterns and the Pythagorean identity. Excellent work! You've successfully proven a trigonometric identity using algebraic techniques. Here's what we learned: Pattern Recognition: Identifying algebraic structures like (a - b)(a + b) is key to proving identities efficiently. Difference of Squares: The formula (a - b)(a + b) = a² - b² is a powerful tool that works with any expressions, including trigonometric functions. Pythagorean Identity: The fundamental identity sin²x + cos²x = 1 can be rearranged to substitute expressions. Here we used sin²x = 1 - cos²x. Strategic Approach: Starting with the more complex side (left) and simplifying toward the simpler side (right) is often the most effective strategy. Verification: Using multiple approaches (algebraic manipulation + identity confirmation) strengthens understanding and validates the proof. This systematic approach combining algebra and trigonometric identities will serve you well in proving more complex identities. Always look for patterns and consider which fundamental identities might help simplify expressions!

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