2. Prove that cos (x + y) + cos (x − y) = 2 cos x cos y.

6. By the method of Example 1, § 72, show that

7. By the method of Example 2, § 72, show that

15. The so-called "method of offsets" for laying out a circular track is illustrated in the adjoining figure. The track OAB is tangent at O to OB', and the distances OA', A'B', A'A, CB, are easily shown to be as marked in the figure, where a/2 = ZAOA' is half the angle at the center subtended by a 100 ft. chord. In practice, the line OA'B' is run, and A' and B' marked. Show that B'B, the distance actually to be laid off from B', is

B'BA'A + CB = 200 sin a cos (a/2).

> 100 cos (3 a/2)

C 100 sin(3/2)B

A

100 sin a/2

is satisfied when any number whatever is substituted for x; we say it is satisfied by all values of x. The equation

(x2-1)/(x-1)=x+1

-

is satisfied by all values of x except x= = 1. The student may verify both of these statements for x=-2, 1, 1/2, 3/2, 2, etc. Similarly, the equation (1+x)(1 − 1/x) = (x − 1)(1+1x) is satisfied by every value of x except x=0. The equation sin2 x + cos2 x = 1 is satisfied by every value of x; and the equation tan x cos x = sin x is satisfied by every value of x except x=an odd multiple of ±90°. These are examples of identities: Two expressions involving an unknown letter are said to be identically equal, or simply, identical, if they have the same value for every value of the unknown for which both are defined.* An equation whose sides are identically equal is called an identity.

74. Elementary Identities. An identity is to be regarded as a declaration to be proved: thus

cos 2x = (cosx + sin x) (cos x - sin x)

declares that for every angle x the cosine of twice that angle is equal to the product of the sum and difference of its cosine and its sine; this was proved in § 68. Among other identities

*The trigonometric functions sinx and cos x are defined for every value of x; tanx and secx, however, are not defined for x = any odd multiple of 90°, while ctnx and cscx are not defined when x = any even multiple of 90°. See p. 61. It is assumed that values of the unknown exist for which both sides are defined. A similar definition holds for identities in several variables.

that have been established are the Pythagorean relations, § 10, p. 16, the reciprocal relations, etc., § 6, p. 9.

75. Identities in Two Variables. In Chapter VI we found: sin (x+y)= sin x cos y + cos x sin

cos (x + y)

for all values of x and y.

y,

)= cos x cos y — sin x sin y,

These are identities in two variables.

76. Illustrative Example. The truth of an identity is usually established by reducing both sides, either to the same expression, or to two expressions which are known to be identical. Example 1. Prove that 1 - sin 0 = cos2 0/(1 + sin @) is an identity. The right-hand side is not defined when 1 + sin 0 = 0. The left-hand side has a value for every value of 0. We are to show, then, that the two sides of the equation have the same value for every value of @ except those that make 1 + sin 0 = 0; i.e. except when ≈ 270°. To prove this we reduce the right-hand side to the left-hand side. Replace cos20 by 1 — sin2 0, then cos2 0/ (1 + sin 0) = (1 − sin2 0) /(1+ sin @). Dividing the numerator by the denominator we obtain 1 sin 0, which is the left-hand side of the given equation. This division is permissible if 1 + sin 0 ‡ 0.

EXERCISES XXX. - TRIGONOMETRIC IDENTITIES

Prove the truth of the following identities and state in each case the exceptional values of the variables, if any, for which one or both of the two sides are undefined.

16. sin 3x = sin x (3 – 4 sin2 x) = sin x (2 cos x − 1) (2 cos x + 1).

3) = cos x (1 - 2 sin x) (1 + 2 sin x).

18. 1 + sin x- cos 2 x tan x(cos x + sin 2 x).

19. (1+ cos 2 x) tan x = sin 2 x.

20. [sin(x/2)

21. [sin(x/2)

· cos (x/2)]2 = 1 + sin x = 2 cos2 (45° — x/2).

22. sec (45° - x/2) sec (45° + x/2) = 2 sec x.

Prove that the following expressions are reciprocals :

23. secx+tan x and sec x tan x.

[NOTE. Two numbers are reciprocals if and only if their product is + 1; in Ex. 23 we must prove that (sec x + tan x) (sec x tan x)= 1. Just as in any other identity, values of x for which either side is meaningless are excluded. Values of x that make either of the given expressions vanish must be excluded; thus, in Ex. 24, 1 - sin x = : 0 when x 90°; and in Ex. 25, 1+ cos x = 0 when x

270°.]

77. Conditional Equations. In the exercises of the preceding list it was frequently necessary to determine the values of x which would make a certain expression vanish. Thus in Exs. 24-25, the equations 1 - sin x = 0, 1+ cos x = 0, etc., were considered. These are not identities, since it is clear that there are values of x in each case for which the left-hand side is defined and for which that side is different from zero.

An equation in a which is not satisfied by all values of x for which each side is defined is called a conditional equation, or, when no ambiguity can arise, simply an equation.

Examples of conditional equations that are quite familiar are:

(a) x2 - 5x+6= 0,

which is satisfied by two and only two values of x:

(b) 8x8 - 12 x2 + 6x = 1,

which is satisfied by one and only one value of x:

(c) 4 cost x sin2 2 x = 2,

which is satisfied by x = - 45°, 135°, 225° 315°, and all angles congruent to any one of these. This last equation, therefore, has an infinite number* of solutions, but nevertheless it is not an identity, since there exist values of x for which the two sides have two definite values that are different.

* There are an infinite number of things in a class of things, if, when you have counted out as many as you please, still others remain.

The purpose of what follows is to show how to find all the solutions of certain simple forms of equations containing trigonometric functions of an unknown angle.

Any equation that is not an identity is to be regarded, not as a declaration to be proved, but rather as a question to be investigated and answered. Thus the equation,

implies the question, "Are there any values of x which make 823-12x2+6x equal to 1?" and the direction, "If so, find all of them.” This is the meaning of the direction, "Solve the equation." This point of view is very important.

78. Illustrative Examples. The simplest trigonometric equations are of the form

The method of solving such equations is illustrated in the examples that follow:

Example 1. Solve the equation sin x = 1/2.

We know that x = 30° is a solution, and that any angle congruent to 30° must be a solution: x = 30°, 750°, - 330°, etc. All these angles are included in the statement x≈ 30°; but there are still other solutions, since we know that the sine of an angle is also the sine of its supplement ; the supplement of 30°, or 150°, must therefore be a solution, and hence all angles x such that x 150° are solutions. We shall show presently

that there are no others.

Example 2. Solve the equation cos x = - 4/5.

The value x = 143° 8′ (approximately) is a solution, as may be verified by a table of cosines; hence other solutions are x 143° 8'. Are there still others?

Example 3. Solve the equation tan x = 1/3.

From the tables an approximate solution is found to be x = 18° 26'. Hence other solutions are x 18° 26'. Are there still others?

79. General Principles.

A general method of solving such

equations depends upon the following theorems :

* We shall consider only real values of x, since in elementary work the trigonometric functions are not defined for imaginary values of the angle.