Overview
This lecture covers key strategies and step-by-step examples for verifying trigonometric identities using algebraic manipulation and fundamental trigonometric relationships.
Techniques for Verifying Trig Identities
- Convert expressions to sine and cosine to simplify.
- Combine terms with common denominators to merge two fractions into one.
- Split one term into multiple terms by distributing or separating fractions if needed.
- Convert between division and multiplication as appropriate.
- Multiply by the conjugate when indicated by expressions like "1 ± cosine" in denominators.
- Apply factoring or FOIL (First, Outside, Inside, Last) when applicable, especially with differences of squares or trinomials.
Essential Trigonometric Identities
- ( \sin^2 x + \cos^2 x = 1 )
- ( 1 + \tan^2 x = \sec^2 x )
- ( 1 + \cot^2 x = \csc^2 x )
- ( \tan x = \frac{\sin x}{\cos x} )
- ( \cot x = \frac{\cos x}{\sin x} )
- ( \tan x = \frac{1}{\cot x} )
- ( \sec x = \frac{1}{\cos x} )
- ( \csc x = \frac{1}{\sin x} )
Example Problems & Solutions
Problem 1: ( \sin x \cdot \sec x = \tan x )
- Rewrite secant as ( 1/\cos x ).
- Simplify ( \sin x / \cos x = \tan x ).
Problem 2: ( \tan^2 x \cdot \cot^2 x = 1 )
- ( \tan^2 x = (\sin x/\cos x)^2 ), ( \cot^2 x = (\cos x/\sin x)^2 ).
- Multiply and simplify to 1.
Problem 3: ( \cot x \cdot \sec x \cdot \sin x = 1 )
- Change all functions to sines and cosines, cancel terms, and simplify to 1.
Problem 4: ( \frac{\cos x \cdot \sec x}{\cot x} = \tan x )
- Cancel ( \cos x ) and ( \sec x ), recognize ( 1/\cot x = \tan x ).
Problem 5: ( \sin x \cdot \tan x = \frac{1 - \cos^2 x}{\cos x} )
- Replace ( \tan x ) with ( \sin x/\cos x ), ( \sin^2 x = 1 - \cos^2 x ).
Problem 6: ( \cos^2 x - \sin^2 x = 1 - 2\sin^2 x )
- Express ( \cos^2 x ) as ( 1 - \sin^2 x ), combine like terms.
Problem 7: ( \sin x \tan x + \cos x = \sec x )
- Convert all terms to sines and cosines, combine fractions with common denominators, use identities.
Problem 8: ( \sec x - \cos x = \tan x \cdot \sin x )
- Write both terms over a common denominator, use ( 1 - \cos^2 x = \sin^2 x ), simplify the expression.
Key Terms & Definitions
- Sine (sin) — Ratio of opposite side to hypotenuse in a right triangle.
- Cosine (cos) — Ratio of adjacent side to hypotenuse.
- Tangent (tan) — Ratio of sine to cosine.
- Cotangent (cot) — Reciprocal of tangent; cos/sin.
- Secant (sec) — Reciprocal of cosine; 1/cos.
- Cosecant (csc) — Reciprocal of sine; 1/sin.
- Identity — An equation true for all values in its domain.
- Conjugate — Expression with the sign between terms reversed, used to rationalize denominators.
Action Items / Next Steps
- Memorize the core trigonometric identities listed above.
- For each example, practice rewriting the steps in your own notebook.
- Pause and attempt each problem before viewing the solution.