(1) sin (A+B) + sin (A − B) = 2 sin A. cos B.

(2) sin (A+B) - sin (A - B)=2 cos A. sin B.

(3) cos (A+B) + cos (A (4) cos (A-B) - cos (A

(5)

− B)=2 cos A.

cos B. + B) = 2 sin A. sin B.

sin (A+B) + sin (A – B) ̧
cos (A+B)+cos (A – B)

(6) tan a+tan ß:

=

(7) tan a- tan B=

(8) cota +tan ß=

(9) cot a-tan ß=

(10) tan a + cot ß=

=tan A.

sin (a+B) cos a. cos B

sin (a-B) cos a. cos B

cos (a - B)

tan - tan

sin (0-0)

(24)

tan2 a tan2 B 1-tan2 a α. tan2 B

sin (a + B). sin (a - ẞ)

cos2 a. cos2 B

cos (a+B). cos (a -B)

=

sin2 a. cos2 B

=tan (a + ß) . tan (a – ß).

sin (a +ẞ). sin (a — ß)=sin2 a — sin2 ß=cos2 ß - cos2 a.

(25) cos (a + B). cos (a-B)=cos2 a - sin2 ß=cos2B-sin2 a.

(31)

(32)

(33)

sin (0+). cos 0—cos (0 +$). sin 0=sin 0.
sin (0-4). cos + cos (8-4). sin &=sin 0.

cos (8+). cos + sin (0+). sin 0=cos p.

(40) sin nA.cos A + cos nA. sin A=sin (n + 1) A.

(41) cos (n-1) A. cos A - sin (n − 1) A. sin A=cos nA. (42) sin nA. cos (n − 1) A − cos nA. sin (n − 1) A=sin A.

(43) cos (n-1) A. cos (n + 1) A-sin (n - 1) A. sin (n + 1) A

should prove the second in a similar manner. The third and fourth follow at once from the first two by putting 45° for B.

Example. Το prove tan (A + B)

=

tan A+tan B
1-tan A. tan B*

(i) By using the results of Art. 154, we have

tan (A+B)=

sin (A+B) sin A. cos B+cos A. sin B
cos (A+B) cos A. cos B- sin A. sin B'

=

Divide the numerator and the denominator of this fraction each by cos A. cos B, and we get

(ii)** By Geometry. Make the construction of page 118;

(1) If tan A= and tan B=1, prove that tan (A+B)=§, and

No3'

, prove that

N

tan (A+B)=2+√3.

(3) Prove that tan 150-2-√3.

(4) If tan A=§_and_tan B=, prove that tan(A+B)=1.

What is (A+B) in this case?

(15) If tan a=m and tan ß=n, prove that

(16) Iftan a = (a + 1) and tan ẞ = (a−1), then 2 cot (a – B) = a2.

(17) If a+B+y=90°, then