Proof. Express cot a and tan ẞ in terms of sine and cosine;

then

If α + β + γ

180°, prove the identities in the following problems.

17. sin a + sin 6 + sin y == 4 sin (a + B) cosa cos B. Proof. sin a + sin ß

Now y = 180°

=

2 sin 1⁄2 (a + ẞ) cos 1⁄2 (a — ß), by [25].

(a + B).

γ sin [180° - (a + B)]

sin Y

=

sin (a + B) = 2 sin 1 (a + ẞ) cos 1⁄2 (a + ẞ), by [19].

sin (a + B), Art. 45.

..

B) + cos(a + B)

B) + 2 sin (a + ẞ) cos (a + B)
1⁄2

B) + cos 1⁄2 (a + B)].

2 cos 1⁄2 [1⁄2 (α – ß) + 1⁄2 (α + ß)] cos 1⁄2 [1⁄2 (α — ß) − 1 (a + ß)]

2 cosa cos B.

sin a + sin ẞ + sin y

2 sin (a + B) 2 cosa cos B
4 sin (a + B) cosa cos B.
(a + ẞ) sin a sin ẞ + 1.
-4 cos a cos ẞ cos y — 1.
4 sin a sin ẞ sin y.

74. To change the product of functions of angles into the sum of functions.

From Art. 65,

(d)

cos (a B) = cos a cos ẞ + sin a sin ß.

Adding (a) and (b), sin (a + ß) + sin (a — ẞ) = 2 sin a cos B.

[29]

sin (a + ẞ) + 1⁄2 sin (a — ß).

.. sin a cos ẞ = Subtracting (b) from (a), sin (a+B) — sin (a-ẞ) = 2 cos a sin ß. [30] .. cos a sin ẞ = sin (a + ẞ) — 1 sin (a — ß). Adding (c) and (d), cos (a + ß) + cos (a [31] .. cos a cos ẞ = Subtracting (d) from (c), cos (a + B) .. sin a sin ẞ

[32]

B) 2 cos a cos B. cos (a + B) + 1 cos (a — ẞ).

cos (a — ß)

2 sin a sin ẞ.

· 1/2 cos (a + B) + 1⁄2 cos (a — ẞ).

EXERCISES

In the following, when an identity is to be proved, transform one member into the other by the application of methods previously suggested.

19. sin2 a tan a + cos2 a cota + 2 sin a cos a = sec a csc a.

Suggestion. Square the first equation and collect the terms in x2 and y2.

This gives the square of x sin 0 + y cos 0

0. Then tan 0

y
X

From

W2

this find sin ✪ and cos ✪ and substitute in the second equation.
25. Given P W sin 0, P cos 0 = x sin 0; show that P2
26. Given a sin 0 + b cos 0 = c, and a cos 0 — b sin 0

=

x2.

d; eliminate 0.

Ans. a2 + b2 c2 + d2.

sin-1 x √3
X

Suggestion. Find the sine of the first two angles and then combine with

the third.

29. Given sin−1 2 x

sin-1 x; find x. Ans. x = 0 or ±1⁄2.

47. If 2 tan 2 α = tan 2 ẞ sin 2 ß, find the relation between the tangents of a and B.

√1 + sin 2 a

cot a.

49. In any triangle with angles a, ẞ, y, and opposite sides a, b, c, respectively, show that

= a cos y + c cos a.

b

α

cos B
+ +
b

COS Y

a2 + b2 + c2

с

2 abc