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Since we had no 4 additional restrictions on t the

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Unformatted text preview: starting with one side of the equation and trying to transform it into the other, we will start with the identity we proved in number 3 of Example 10.4.3 and manipulate it into the identity we are asked to prove. The 2 tan( identity we are asked to start with is sin(2θ) = 1+tan2θ) ) . If we are to use this to derive an (θ identity for tan θ 2 , it seems reasonable to proceed by replacing each occurrence of θ with sin 2 θ 2 2 tan = θ 2 θ 2 1 + tan2 2 tan sin(θ) = 1 + tan2 θ 2 θ 2 θ 2 We now have the sin(θ) we need, but we somehow need to get a factor of 1 + cos(θ) involved. θ θ To get cosines involved, recall that 1 + tan2 2 = sec2 2 . We continue to manipulate our given identity by converting secants to cosines and using a power reduction formula sin(θ) = θ 2 2 tan 1 + tan2 sec2 θ 2 θ 2 sin(θ) = 2 tan θ 2 sin(θ) = 2 tan θ 2 θ 2 sin(θ) = 2 tan sin(θ) = tan tan θ 2 = θ 2 cos2 θ 2 1 + cos 2 2 θ 2 (1 + cos(θ)) sin(θ) 1 + cos(θ) Our next batch of identities, the Product to Sum Formulas,3 are easily verified by expanding each of the right hand sides in accordance with Theorem 10.16 and as you should expect by now we leave the details as exercises. They are of particular use in Calculus, and we list them here for reference. 3 These are also known as the Prosthaphaeresis Formulas and have a rich history. The authors recommend that you conduct some research on them as your schedule allows. 666 Foundations of Trigonometry Theorem 10.20. Product to...
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