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Lecture Notes Trigonometric Identities 1 Sample Problems page 1 Prove each of the following identities. 1. tan x sin x + cos x = sec x 2. 1 1 + tan x = tan x sin x cos x sin x cos2 x = sin3 x + 1 + sin cos = 2 cos 8. 1 2 cos2 x = tan2 x 1 tan2 x + 1 cos x 1 sin x 2 cos2 x 9. sec x + tan x = 10. sin4 x 11. (sin x 3. sin x 4. cos 1 + sin cos4 x = 1 cos x)2 + (sin x + cos x)2 = 2 cos x 5. 1 sin x sin4 x 6. sin2 x 7. 1 cos x = 2 tan x 1 + sin x sin2 x + 4 sin x + 3 3 + sin x 12. = 2x cos 1 sin x 13. cos x 1 sin x tan x = sec x 1 + sin x cos2 x cos4 x =1 cos2 x cos x sin x = cos x 1 + sin x 14. tan2 x + 1 + tan x sec x = Practice Problems Prove each of the following identities. 1. tan x + 2. cos x 1 …show more content…

sin4 x cos4 x = 1 Solution: LHS = sin4 x cos4 x = sin2 x = 1 sin2 x cos2 x = 1 11. (sin x cos x)2 + (sin x + cos x)2 = 2 Solution: cos2 x = sin2 x + cos2 x sin2 x cos2 x cos2 x cos2 x = 1 2 cos2 x = RHS 2 LHS = (sin x cos x)2 + (sin x + cos x)2 = sin2 x + cos2 x 2 sin x cos x + sin2 x + cos2 x + 2 sin x cos x = 2 sin2 x + 2 cos2 x = 2 sin2 x + cos2 x = 2 1 = 2 = RHS c copyright Hidegkuti, Powell, 2009 Last revised: March 16, 2011 Lecture Notes Trigonometric Identities 1 page 4 12. sin2 x + 4 sin x + 3 3 + sin x = 2x cos 1 sin x Solution: LHS = sin2 x + 4 sin x + 3 (sin x + 1) (sin x + 3) (sin x + 1) (sin x + 3) = = = 2 cos2 x (1 + sin x) (1 sin x) 1 sin x sin x + 3 = = RHS 1 sin x 13. cos x 1 sin x Solution: tan x = sec x cos x cos x sin x cos2 x sin x (1 sin x) cos2 x sin x + sin2 x tan x = = = LHS = 1 sin x 1 sin x cos x cos x (1 sin x) cos x (1 sin x) 2 2 cos x + sin x sin x 1 sin x 1 = = = = RHS cos x (1 sin x) cos x (1 sin x) cos x 1 + sin x 14. tan2 x + 1 + tan x sec x = cos2 x Solution: sin2 x sin x 1 sin2 x cos2 x sin x LHS = tan x + 1 + tan x sec x = +1+ = + + 2x 2x 2x cos cos x cos x cos cos cos2 x 2 2 1 + sin x sin x + cos x + sin x = = RHS = 2x cos cos2 x 2 c copyright

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