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Section 6.1: Verifying Trigonometric Identities Lecture 28 – Typeset by FoilTEX – Outline • Definition Identities of Trigonometric • Fundamental Identities • Examples (x,y) t Trigonometric Identities • Let f (t), g(t) be functions which involve some (or all) of the six basic trigonometric functions: sin(t), cos(t), tan(t), csc(t), sec(t), cot(t). • A trigonometric identity is an equation of the form f (t) = g(t) which holds true for all values of t in the domain of f and g. ∗ tan(t) = sin(t) cos(t) is a trigonometric identity. ∗ sin(t) = cos(t) is not a trigonometric identity. Fundamental Identities • Reciprocal Identities: 1 csc(t) = sin(t) 1 sec(t) = cos(t) 1 cot(t) = tan(t) • Tangent and Cotangent Identities: tan(t) = sin(t) cos(t) cot(t) = cos(t) sin(t) Fundamental Identities • Pythagorean Identities: ∗ sin2(t) + cos2(t) = 1 ∗ 1 + tan2(t) = sec2(t) ∗ 1 + cot2(t) = csc2(t) Verifying Identities • There are a couple of ways to verify (or prove) a trigonometric identity f (t) = g(t). 1) Use the fundamental identities and algebra to transform the left hand side into the right hand side (or vice versa). 2) First use fundamental identities and algebra to transform the left hand side into some equivalent expression h(t). Then use fundamental identities and algebra to transform the right hand side into h(t). Example • Verify the identity sec(t) = sin(t)(tan(t) + cot(t)). Example • Verify the identity 1 + sin(x) cos(x) = . 1 − sin(x) cos(x) Example • Verify the identity (tan θ − sec θ)2 = 1 − sin θ . 1 + sin θ