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Handy-Dandy Trig Identities for Calculus June 2, 2015 Instructor: Maxx Cho Math 141, Section 0101 Name: 1. The pythagorean identity: sin2 x + cos2 x = 1 Divide both sides by sin2 x to get: 1 + cot2 x = csc2 x Divide both sides by cos2 x to get: tan2 x + 1 = sec2 x 2. The sum/difference formulas: sin A ± B = sin A cos B ± cos A sin B cos A ± B = cos A cos B ∓ sin A sin B Note the difference between ± and ∓. 3. In the angle sum formulas in (2), you can set A = B and simplify to get the double-angle formulas: sin 2A = 2 sin A cos A cos 2A = cos2 A − sin2 A In the cosine double-angle identity, you can substitute with the pythagorean identity in (1) and simplify to get: cos 2A = 2 cos2 A − 1 cos 2A = 1 − 2 sin2 A 4. Take the last two cosine double angle formulas in (3), and solve them for cos2 A and sin2 A to get the power-reduction formulas cos 2A + 1 cos2 A = 2 1 − cos 2A sin2 A = 2 sin x In (2), (3), and (4), you can use the fact that tan x = cos x to derive the analogous formulas for tan x as well. 5. Also, don’t forget some basic facts. (1) sin x is an odd function. That means sin(−x) = − sin x. (2) cos x is an even function. That means cos(−x) = + cos x. (3) Since csc x = sin1 x , it is also odd. (4) Since sec x = cos1 x , it is also even. sin x (5) Since tan x = cos x , it is odd. So is cot x. 1