Proof of the sum formulas
Proof. Let the straight line AB revolve to the point C and sweep out the
angle , and let it continue to D and sweep out the angle β;
draw DE perpendicular to AB.
Note: The value of a trigonometric function is a number, namely the number that represents the ratio of two lengths. Throughout the proof, then, we will consider AE and DA not only as lengths, but also as the numbers that are their measures. Therefore the usual properties of arithmetic will apply.
Draw DF perpendicular to AC,
draw FG perpendicular to AB,
and draw FH perpendicular to ED.
Then angle HDF is equal to angle .
therefore angle HFA is equal to angle .
And by the construction, angle DFH is the complement of angle HFA;
therefore angle HDF (the complement of DFH) is also equal to angle .
And on both dividing and multiplying by AF and FD
This is what we wanted to prove.
The difference formulas can be proved from the sum formulas, by replacing +β with +(−β), and using these identities:
cos (−β) = cos β
sin (−β) = −sin β.
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