701 7.3 Sum and Difference Identities Now we write sin1A - B2 as sin3A + 1-B24 and use the identity just found for sin1A + B2. sin1A - B2 = sin3A + 1-B24 Definition of subtraction = sin A cos1-B2 + cos A sin1-B2 Sine sum identity sin1A −B2 =sin A cos B −cos A sin B Even-odd identities Sine of a Sum or Difference sin1A +B2 =sin A cos B +cos A sin B sin1A −B2 =sin A cos B −cos A sin B We can derive the identity for tan1A + B2 as follows. tan1A + B2 = sin1A + B2 cos1A + B2 Fundamental identity = sin A cos B + cos A sin B cos A cos B - sin A sin B Sum identities = sin A cos B + cos A sin B 1 cos A cos B - sin A sin B 1 # 1 cos A cos B 1 cos A cos B = sin A cos B cos A cos B + cos A sin B cos A cos B cos A cos B cos A cos B - sin A sin B cos A cos B = sin A cos A + sin B cos B 1 - sin A cos A # sin B cos B Simplify. tan1A +B2 = tan A +tan B 1 −tan A tan B sin u cos u = tan u Multiply by 1, where 1 = 1 cos A cos B 1 cos A cos B . Multiply numerators. Multiply denominators. We express this result in terms of the tangent function. Tangent of a Sum or Difference tan1A +B2 = tan A +tan B 1 −tan A tan B tan1A −B2 = tan A −tan B 1 +tan A tan B We can replace B with -B and use the fact that tan1-B2 = -tan B to obtain the identity for the tangent of the difference of two angles, as seen below.
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