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Math 130 Verifying Trigonometric Identities Definitions: sin θ = y r cosθ = x r tan θ = y x cot θ = x y csc θ = r y sec θ = r x r y Reciprocal identities: csc θ = 1 1 1 , sec θ = , cot θ = sin θ cosθ tan θ θ x Negative angle identities: sin( −θ ) = − sin θ ,cos( −θ ) = cosθ , tan( −θ ) = − tan θ r b θ a −θ −b r Identities: sin x cos x → cot x = cos x sin x 2 2 2 2 2 2 2) sin x + cos x = 1 → tan x + 1 = sec x , 1 + cot x = csc x 1) tan x = TO DO • Start with more complex side (more to work with) • Always look for forms of known identities that you can apply (e.g. 1 − sin 2 x = cos2 x ) • Multiply denominators according to table below: Denominator Multiply by 1 ± sin x 1 m sin x 1 m sin x 1 ± cos x 1 m cos x 1 m cos x sec x ± tan x sec x m tan x sec x m tan x sec x ± 1 sec x m 1 sec x m 1 csc x ± cot x csc x m cot x csc x m cot x csc x ± 1 csc x m 1 csc x m 1 • Multiply denominators of by • Convert tan x,cot x,csc x,sec x → sin x,cos x expressions then algebra, typically clearing fractions. Multiply by conjugate only if it gets you a form equivalent to one of the Pythag. identities. Otherwise convert to sin x,cos x .