Plane trigonometry, by S.L. Loney.

[Exs. XV.] TANGENT OF THE SUM OF TWO ANGLES. 99
Prove that
0. 70 30. 110
5. sin ssmn -+ sin- sinm -= sin 20 sin 50.
2 ~ 2 2
0 90 5O
6.os cos2cos - cos 30 cos -= sin 50 sin.
2 2 2
7. sin A sin (A + 2B) - sin B sin (B +2A) = sin (A - B) sin (A + B).
8. (sin 3A + sin A) sin A + (cos 3A - cos A) cos A =0.
2 sin (A - C) cos C - sin (A - 2C) sin A
2 sin (B - C) cos C- sin (B - 2C) sin B'
sin A sin 2A + sin 3A sin 6A + sin 4A sin 13A
10. = tan 9A.
sin A cos 2A + sin 3A cos 6A + sin 4A cos 13A 9.
cos 2A cos 3A - cos 2A cos 7A + cos A cos 10A
11. = cot 6A cot 5A.
sin 4A sin 3A - sin 2A sin 5A + sin 4A sin 7A A
12. cos (36~ - A) cos (36~ + A) + cos (54~ + A) cos (54~ - A) =cos 2A.
13. cos A sin (B - C) + cos B sin (C - A)+cos C sin (A - B) =0.
1
14. sin (45 +A) sin (45 -A) = cos2A.
15. versin (A + B) versin (A - B) = (cos A - cos B)2.
16. sin ( -y) cos (a - 6) + sin (y - a) cos ( - ) + si (a - ) cos ( - ) = 0.
1 c r 97r 37r 57r -
17. 2cos 1c os +cos +cos=0.
tan A + tan B
98. To prove that tan (A + B) = tan A +ta and
I -tan A tan B'- tan
that tan (A - B)1 = tan A-tan B
1 + tan A tan B
By Art. 88, we have, for all values of A and B,
B sin (A + B) sin A cos B + cos A sin B
cos (A + B) cos A cos B - sin A sin B
sin A sin B
oA cosA c B
sin A sin B' by dividing both
cos A cos B
numerator and denominator by cos A cos B.
tan A + tan B
tan (A+B)== - A —
-tan A tan B
7-2