Plane trigonometry, by S.L. Loney.

ADDITION AND SUBTRACTION FORMULE.

89

Hence the formulae of Art. 88 are true if either A or B be increased
by 90~, i.e. they are true if the component angles lie between 0~ and
180~.
Similarly, by putting A2=90~+A1, we can prove the truth of the
theorems when either or both of the component angles have values
between 0~ and 270~.
By proceeding in this way we see that the theorems are true universally.
90. Theorem. To prove that
sin (A - B)= sin A cos B - cos A sin B,
and cos (A - B) = cos A cos B + sin A sin B.
Let the revolving line starting from the initial line
OA trace out the angle
AOB (= A) and then revolving in the opposite direction, trace out the angle N.,.R
BOG, whose magnitude is - C
B. The angle A OC is therefore A -B.
Take a point P in the d Q Mt A
final position of the revolving line, and draw PM and PA perpendicular to OA and
OB respectively; from N draw ATQ and NR perpendicular
to OA and MP respectively.
The angle RPN= 90~- z PNR= z RNB= z QON=A.
Hence
MP MR-PR QN PR
sin (A - B) = sin AOC = P =
QN ON PR PN
ON OP PN3 OP
= sin A cos B - cos RPN sin B,
so that sin (A - B)= sin A cos B - cos A sin B.