SUMMARY
The equation tan(x) + sec(x) = 1 can be solved by first isolating tan(x) to get tan(x) = 1 - sec(x). Squaring both sides leads to the equation tan^2(x) = 1 - 2sec(x) + sec^2(x). The solution process reveals that sec(x) = 1 is the only valid solution, as sec(x) cannot equal 0. This confirms that the equation has a unique solution at specific angles where sec(x) equals 1.
PREREQUISITES
- Understanding of trigonometric identities, specifically tan(x) and sec(x).
- Familiarity with algebraic manipulation of equations.
- Knowledge of the Pythagorean identity: tan^2(x) + 1 = sec^2(x).
- Ability to solve quadratic equations.
NEXT STEPS
- Study the derivation and implications of the Pythagorean identity in trigonometry.
- Learn how to solve trigonometric equations involving multiple identities.
- Explore the graphical representation of sec(x) and its intersections with y = 1.
- Investigate the periodic nature of trigonometric functions and their solutions.
USEFUL FOR
Students studying trigonometry, educators teaching trigonometric equations, and anyone looking to deepen their understanding of solving trigonometric identities and equations.