☰ Contents · Algebra

Trigonometric identities

Lessons 23 · 1 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
23

Trigonometric identities

Textbook: pp. 117–119
GoalApply methods of proving trigonometric identities and simplifying expressions.
New words
identity · ayniyatproving an identity · ayniyatni isbotlashadmissible values · joiz qiymatlarsimplifying · soddalashtirish
Explanation

An identity is an equality that is true for all admissible values of the angle, i.e. all values for which both sides make sense. The main ways to prove an identity: 1) transform the left side into the right side (or vice versa); 2) show that the difference of the two sides is zero; 3) bring both sides to the same expression. The tools are sin²α + cos²α = 1, tan α = sin α : cos α, cot α = cos α : sin α, 1 + tan²α = 1 : cos²α and 1 + cot²α = 1 : sin²α. A useful move is to write tan and cot through sin and cos and use a common denominator. Simplifying an expression uses the same formulas, helped by the short multiplication formulas (a ± b)² and a² − b². Unless asked, we do not search for admissible values, but we keep in mind that denominators must not be zero.

Worked examples
sin α cos α (tan α + cot α) = 1: sin α cos α (sin α : cos α + cos α : sin α) = sin²α + cos²α = 1.
tan α + cot α = 1 : (sin α cos α): tan α + cot α = sin α : cos α + cos α : sin α = (sin²α + cos²α) : (sin α cos α) = 1 : (sin α cos α).
Class activity

“Identity fixer”: a deliberately flawed proof is shown on the board; pairs find the error and rewrite the proof correctly.

Practice
1
Simplify (sin α + cos α)² − 2 sin α cos α.
2
Prove that cos α · tan α = sin α.
3
Compute 3 sin² 20° + 3 cos² 20°.
4
Why must sin α ≠ 0 in the identity 1 + cot²α = 1 : sin²α?