Below, some proofs are presented of trigonometric identities.
The proofs do not use any trigonometric formula or rule, what makes them quite special.
Identity Nr. 1
See figure 1. below.
Pictured are 4 squares.
LD1 = arctan(
LB1 = arctan(1 )
2. MC = 3.ME
3. AD = 3.AE
(remember: 1800 - LB1 = 1800 - (LC1+LE1)
This concludes proof A.
See figure 2. below:
Note: EF = 5, because of theorem of Pythagoras.
DADG is the rotation of DABE over 900= LGAE.
Now observe polygon AEFG.
AG = AE
LGAD + LDAF = 450
Identity Nr. 2
See figure 3. below:
M is the center of a circle with radius = 1.
Points A,B,C,D are located on the circle at the indicated angles.
Chord DC is extended by CE = BC.
Now DCBE is equilateral because:
LECB = 600 ....so
LCBE = LCEB = 600....so
BE = BC = AB
DE = DA ............and since
CE + DC = DE
AB + DC = AD
2sin(200 )+ 2sin(400 ) = 2sin(800 )........(see appendix 2)
sin(200 )+ sin(400 ) = sin(800 )
Appendix 1 (measuring angles by arcs)
See figure 4. below:
By:...... arc AB = 800 we mean : LAMB = 800
MA = MP
LM1 = 2*LP1
LP1 = 0.5*LM1 = 0.5* arc AC.....2
Appendix 2 (sine and chord relation)
See figure 5. below:
x = 2 · sin