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Below we summarize all of the sum and diﬀerence formulas for cosine, sine and tangent.
Theorem 10.16. Sum and Diﬀerence Identities: For all applicable angles α and β ,
• cos(α ± β ) = cos(α) cos(β ) sin(α) sin(β ) • sin(α ± β ) = sin(α) cos(β ) ± cos(α) sin(β )
• tan(α ± β ) = tan(α) ± tan(β )
1 tan(α) tan(β ) In the statement of Theorem 10.16, we have combined the cases for the sum ‘+’ and diﬀerence ‘−’
of angles into one formula. The convention here is that if you want the formula for the sum ‘+’ of 10.4 Trigonometric Identities 661 two angles, you use the top sign in the formula; for the diﬀerence, ‘−’, use the bottom sign. For
example,
tan(α) − tan(β )
tan(α − β ) =
1 + tan(α) tan(β )
If we specialize the sum formulas in Theorem 10.16 to the case when α = β , we obtain the following
‘Double Angle’ Identities.
Theorem 10.17. Double Angle Identities: For all angles θ, 2
2 cos (θ) − sin (θ) 2 cos2 (θ) − 1
• cos(2θ)...

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