Pythagorean and reciprocal relationships

Pythagorean and reciprocal trigonometric relationships

Connect the unit-circle equation to the fundamental Pythagorean identity and its tangent–secant and cotangent–cosecant forms. This precalculus practice develops the reasoning behind the formulas, with particular attention to denominator restrictions and the signs of square roots.

Work through ratio reconstruction, quadrant-based choices, reciprocal relationships, and short simplifications. Conceptual and error-analysis questions ask when a calculation is legal, what information a squared value loses, and why an expression can remain undefined even after its algebraic form becomes simpler. Multiselect items require checking each proposed statement rather than choosing a familiar-looking formula.

Prerequisites are exact unit-circle values, the six trigonometric definitions, elementary algebra and interval or quadrant notation. Begin with a sketch or sign table and use exact arithmetic. The authored practice moves from derivation to calculations and then to counterexamples and domain-sensitive justification; shuffled questions still stand alone.

Worked explanations show how the identity is obtained, how the quadrant selects a sign and how substitution checks a result. This is focused preparation for trigonometric graphs and later identity work. It does not assume general angle-addition formulas or teach unrestricted symbolic cancellation.