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66. To find the values of sin (x − y) and cos (x − y) in terms of the sines and cosines of x and y.

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Let AOB and BOC denote the angles x and y, respectively; then, AOC = x − y.

From any point C in OC draw CA and CB perpendicular to OA and OB; also, draw BD perpendicular to OA, and BE perpendicular to AC produced.

Since EC and BC are perpendicular to OA and OB, the angles BCE and AOB are equal; that is, BCE = x.

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Whence, sin(x − y) = sin x cosy - cosa siny.

BC

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(13)

OD + BE
OD BE
= +
οὐ
Ос ос

OD OD OB

ос ов Ос

= cos x cosy,

= X = sinx sin y.

BE BE BC

ос BC ос

Whence, cos(x − y) = cos x cos y + sin x siny.

(14)

67. The fundamental formulæ of Arts. 65 and 66 are of great importance, and it is necessary to prove that they hold for all values of x and y.

It is obvious that the proof of Art. 65 is not general, for we have assumed in the construction of the figure that a and y are acute angles, and that x+y is < 90°. Also, in Art. 66, we have taken x and y as acute angles, and x>y.

In order to prove the formulæ universally, we will first show that (11) and (12) hold for all values of x and y, and we can then give a general proof of (13) and (14).

68. We will first prove (11) and (12) when x and y are acute, and x + y > 90°.

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Let DOB and BOC denote the angles x and y, respectively; then, DOC=x+y.

From any point C in OC draw CB perpendicular to OB, and CA perpendicular to OD produced; and draw BD and BE perpendicular to OD and AC.

Since EC and BC are perpendicular to OD and OB, the angles BCE and DOB are equal; that is, BCE=x. We then have, by Art. 32,

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Whence, sin(x + y) = sin x cosy + cos x siny.

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

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69. We have thus proved (11) and (12) when x and y are any two acute angles; or, what is the same thing, when they are any two angles in the first quadrant.

Now let a and b be any assigned values of x and y for which (11) and (12) are true; then by Art. 44,

sin [90°+ (a+b)] = cos (a+b)

and,

= cosa cosb-sin a sin b, by (12); (A)

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Hence (A) may be written in the forms,

sin [(90°+a) +b] = sin (90° + a) cos b + cos (90° + a) sin b, sin [a + (90°+b)] = sin a cos (90° +b) + cos a sin (90°+b),

both of which are in accordance with (11).

And (B) may be written in the forms,

cos [(90°+a) +b] = cos (90°+ a) cos bsin (90°+ a) sin b, cọs [a + (90°+b)] = cos a cos (90°+b) — sin a sin (90°+b), both of which are in accordance with (12).

It follows from the above that if (11) and (12) hold for any assigned values of x and y, such as a and b, they also hold when either a or b is increased by 90°.

But they have been proved to hold when both x and y are in the first quadrant; hence they also hold when x is in the second quadrant and y in the first. And since they hold when x is in the second quadrant and y in the first, they also hold when x is in the third quadrant and y in the first; and

so on.

Thus (11) and (12) are proved to hold for any values of x and y whatever, positive or negative.

70. We may now give a general proof of (13) and (14). By (11) and (12), we have

sin(x − y) = sin [x+(− y)]

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

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= cos x cos y + sin x siny (Art. 42).

Hence (13) and (14) hold for all values of x and y, for the above proof depends on formula which have been shown to hold universally.

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Dividing each term of the fraction by cos x cos y,

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Dividing each term of the fraction by sin x sin y,

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72. From (11), (12), (13), and (14), we obtain

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cos (a + b) cos (a - b) = − 2 sin a sin b.

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