COS X cos y = −2 sin † (x + y) sin † (x − y).

(22)

Since cos2 = 1-sin2x, and sin2 = 1 - cos2 x (Art. 19);

we also have

and

cos 2 x = - 1 sin2 x - sin2 x = 1 − 2 sin2 x, (26)

=

cos 2 = cos x-(1- cos x)= 2 cosx-1. (27)

In like manner, from (15) and (17), we have

FUNCTIONS OF x.

75. From (26) and (27), we have

2 sin2 x = 1-ços 2 x, and 2 cos2 x = 1 + cos 2 x. Writing x in place of 2x, and therefore in place of x,

2 sin2 1 x = 1 — cos x, and 2 cos2 ~ = 17 cos x. Dividing by 2, and extracting the square root,

(A)

Multiplying the terms of the fraction under the radical sign first by 1+ cos x, and then by 1 - cos x, we have

And since the cotangent is the reciprocal of the tangent,

Note. The radical in each of the formulæ (30), (31), and (32) is

to be taken as positive or negative according to the quadrant in which the angle is situated (Art. 37).

INVERSE TRIGONOMETRIC FUNCTIONS.

76. The expression sin-1y, called the inverse sine of y, or the anti-sine of y, is used to denote the angle whose sine is equal to y.

Thus the fact that the sine of the angle x is equal to y may be expressed in either of the ways

In like manner, the expression cos 1y signifies the angle whose cosine is equal to y; tan ̄1y, the angle whose tangent is equal to y, etc.

Note. The student must be careful not to confuse this notation with the exponent -1; the -1 power of sin x is expressed (sin x)−1, and not sin-1x.

77. By aid of the principles of Art. 76, any relation involving direct functions may be transformed into one involving inverse functions.

Take for example the formula

sin (x+y)=sin x cos y + cos x sin y.

(A)

Putting sin x = a, and sin y=b, we have

COS X = √1-a2, and cos y = √1 — b2.

Substituting these values in (A),

sin (sin 1a+sin-1b) = a √1 — b2 + b√1 − a2.

Whence by Art. 76,

sin-1a + sin¬1b = sin ̄1 (a √√ 1 — b2 +b √1 — a2).

cos'a + sin2x

Dividing each term of the fraction by cos2 x, we have

1 +

sin2 x cos2x

15. sin(x+y) sin (xy) = sin2x - sin2y.

16. cos(x+y) cos (x − y) = cos2x - sin2y.

1 17. sec2x csc2x = sec2x + csc2x.

18. cos y+cos (120° + y) + cos (120° — y) =

= 0.

19. sin A sin (B − C') + sin B sin (C — A)
+ sin Csin (4-B)=0.

20. cos (A+B) cos (AB) + cos (B+C) cos (B-C)
+cos(C+4) cos (C-A)= cos 2A+cos 2 B+cos 2 C.

By putting x = x + y and y = z in (11) and (12), Art. 65, prove:

23. sin (x + y + z) = sin x cos y cosz + cos x sin y cos z

+cos a cos y sin z − sin

sinx sin y cos z

cos x sin y sin z.