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that is, the cosine of the difference of two arcs, is equal to the rectangle of their cosines, plus the rectangle of their sines.

If we divide Formula (A) by Formula (@), member by member, we have,

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Dividing both terms of the second member by cos a cos b, recollecting that the sine divided by the cosine is equal to the tangent, we find,

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that is, the tangent of the sum of two arcs, is equal to the sum of their tangents, divided by 1 minus the rectangle of their tangents

If, in Formula (), we substitute - b, for b, recollecting that

tan (-6)
b) =

tan b, we have,

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that is, the tangent of the difference of two arcs, is equal to the difference of their tangents, divided by 1 plus the rectangle of their tangents.

In like manner, dividing Formula () by Formula (A), member by member, and reducing, we have,

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and thence, by the substitution of b, for b,

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FUNCTIONS OF DOUBLE ARCS AND HALF ARCS.

66. If, in Formulas (A), (©), (B), and (G), we

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Dividing Equation (A'), first by Equation (4), and then by Equation (3), member by member, we have,

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Taking the reciprocals of both members of the last two

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67. If Formulas (A) and (B) be first added, member to member, and then subtracted, and the same operations be performed upon (C) and (D), we shall obtain,

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=

and then substitute in the above formulas, we obtain,

sin psin q =

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From Formulas (L) and (K), by division, we obtain,

=

sin p - sin q cos (p+q) sin sin p + sin q sin (p+q) cos

=

(p−q) tan (p-q)
(p-q) tan (p+q)

(1.)

That is, the sum of the sines of two arcs is to their dif ference, as the tangent of one half the sum of the arcs is to the tangent of one half their difference.

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all of which give proportions analogous to that deduced from Formula (1).

Since the second members of (6) and (4) are the same, we have,

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That is, the sine of the difference of two arcs is to the difference of the sines as the sum of the sines to the sine of the sum.

All of the preceding formulas may be made homogeneous in terms of R, R being any radius, as explained in Art. 30; or, we may simply introduce R, as a factor, into each term as many times as may be necessary to render all of its terms of the same degree.

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