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Trigonometric Identities:sin2A + cos2A = 1 1 + tan2A = sec2A 1 + cot2A = cosec2A Sum and Difference formula:sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B Product to Sum formula:2sinA cosB = sin(A + B) + sin(A - B) 2sinA sinB = cos(A - B) - cos(A + B) 2cosA sinB = sin(A + B) - sin(A - B) 2cosA cosB = cos(A + B) + cos(A - B) Sum to Product FormulasinC + sinD = 2sin((C + D)/2) cos((C - D)/2) sinC - sinD = 2cos((C + D)/2) sin((C - D)/2) cosC + cosD = 2cos((C + D)/2) cos((C - D)/2) cosC - cosD = 2sin((C + D)/2) sin((C - D)/2) Miscellaneous Formulassin 2A = 2sinA cosA = 2tanA/(1 + tan2A) Proof: sin 2x = 2 sin x cos x Now multiply through by cos x / cos x, giving: 2 sin x cos2x / cos x Which can be written: 2 (sin x / cos x) * cos2x We know that cos = 1 / sec, so, this can be written: 2 (sin x / cos x) * 1 / sec2x We know sin x / cos x = tan x, and sec2x = 1 + tan2x, so, we get: sin 2x = 2 tan x / (1 + tan2x) cos 2A = cos2A - sin2A Proof: We know cos x = 1 / sec x, so, this becomes: (2 / sec2x) - 1 We know sec2x = 1 + tan2x, so: 2 / (1 + tan2 x) - 1 You can them simplify, as follows, by putting it all over a common denominator: 2 / (1 + tan2x) - (1 + tan2x) / (1 + tan2x) = (2 - (1 + tan2x)) / (1 + tan2x) = (1 - tan2x) / (1 + tan2x) So we have: sin 2x = 2 tan x / (1 + tan2x) cos 2x = (1 - tan2x) / (1 + tan2x) sin 3A = 3sinA - 4 sin3A cos 3A = 4cos3A - 3cosA tan 3A = (3tanA - tan3A)/(1 - 3tan2A)
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