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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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