Plane trigonometry, by S.L. Loney.

EXAMPLES. 359
Aliter. Without assuming the series of Art. 294, which is difficult to
remember, the equation (2) may be deduced directly from De Moivre's
Theorem.
For the equation (cos + i sin )7= - 1............................(5),
i.e. cos70+ i sin70= -1,
is clearly satisfied when 0 has either of the values
vr 3r 57r 77r 9rr 11 nd 137r
7, 77, 7. 7 and 7.(6).
7'' 7 ' 7 ' 7-' 7 7 ---.............
Writing c for cos 0 and s for sin 0, the equation (5) on being expanded
by the Binomial Theorem becomes
c7 + 7i6 s - 21c5s2 - 35ic4s3 + 35c3s4 + 21ic2s5 - 7cs6 - is7= - 1.
Equating the real parts on each side, we have
c7 - 21c5 s2 + 35c3 s4 - 7cs + 1 = 0.
Putting s2= 1 - c2, we see that the cosine of each of the angles (6) satisfies
the equation
64c7 - 112c5 + 56c3 - 7c + 1= 0.
But this is equation (2).
# 301. Ex. Find the value of
sec 0 + +sc + sec ( + ) +... to n terms,
sec2 0 + sec2 (+ ) +sec2 (O+ ~) +... to n terms.
From equations (2) and (3) of Art. 296, we know that
n n (n- 12) n (n2- - 12) (n2 3) + ( 1) 2 2n
ne _3 + + 1) 2 2n n
n-I
= (-1) 2 cosnO.............................. (1),
when n is odd,
and that
n2 2(n- 22) I
1 C2+(n2 - 22) C4 +... + (1)2 2n- ( 1)-cosn0......(2),
when n is even,
where in each series c stands for cos 0.