Proof of the sum formulas
angle , and let it continue to D and sweep out the angle draw DE perpendicular to AB.
Draw DF perpendicular to AC, draw FG perpendicular to AB, and draw FH perpendicular to ED. Then angle HDF is equal to angle . For, since the straight line AC crosses the parallel lines HF, AB, it makes the alternate angles equal (Theorem 8); 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 . Now,
And on both dividing and multiplying by AF and FD
Next,
This is what we wanted to prove. The difference formulas can be proved from the sum formulas, by replacing cos (− sin (− Back to Trigonometric identities Please make a donation to keep TheMathPage online. Copyright © 2012 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |