Equations needing substitutions and checking

Trigonometric equations: substitutions and checking solutions

Practise choosing identities that turn a mixed trigonometric equation into a manageable algebraic problem. This set brings together Pythagorean substitutions, double-angle formulas, factoring, radical equations and parameter reasoning. Every answer must satisfy the original equation and its stated interval.

Single-select and multiselect questions distinguish candidate generation from final validation. Detailed explanations show how dividing by a trigonometric factor can lose solutions, how squaring can introduce extraneous roots and how a cancelled denominator remains excluded. Some equations have several solutions and one has none, so counting candidates without checking is not enough.

Suggested solution routine

Record the interval and original exclusions, choose a single trigonometric variable, solve the algebraic equation, then recover all relevant angles. Substitute each candidate into the original expression. Exact radian values are used throughout, and each task stands alone. Revisit quadratic equations, interval solutions and double-angle identities if the corresponding step is difficult. This is progressive practice rather than a diagnostic or mixed assessment.