Plane trigonometry, by S.L. Loney.

92 TRIGONOMETRY. LExs. XIII.]
7. sin (450~ + ) cos (45 - B) + cos (450 + A) sin (45 - B) = cos (A - B).
sin(A-B) sin (B - C) sin (C- A)0
cos Acos cos B cosC cos C cos
9. sin 105~ + cos 105 = cos 45~.
10. sin 75 - sin 15= cos 105~ + cos 15~.
11. cos a cos ( - a) - sin a sin (y - a) = cos a.
12. cos (a + /) cos - cos (P + y) cos a = sin 3 sin (y - a).
13. sin (t + 1) A sin (n - 1) A + cos (n + 1) A cos (n - 1) A= cos 2A.
14. sin (n+1) A sin (n+ 2) A +cos (n +1) A cos (? + 2) A =cos A.
94. From Arts. 88 and 90, we have, for all values of
A and B,
sin (A + B) = sin A cos B + cos A sin B,
and sin (A - B)= sin A cos B - cos A sin B.
Hence, by addition and subtraction, we have
sin (A + B) + sin (A - B) = 2 sin A cos B......(1),
and sin (A + B) - sin (A - B) = 2cos A sin B......(2).
From the same articles we have, for all values of A
and B,
cos (A + B) = cos A cos B - sin A sin B,
and cos (A - B) = cos A cos B + sin A sin B.
Hence, by addition and subtraction, we have
cos (A + B) + cos(A - B) = 2 cos os B......(3),,and cos (A - B) - cos (A + B)= 2 sin A sin B......(4).
Put A + B = C, and A - B = D, so that
0+D C -D2
A = ---, and B = -
2 2