Plane trigonometry, by S.L. Loney.

EXPANSIONS OF SIN nO AND COS nO. 327
Since s- =3, it follows, by the last article, that
tan 1 (+02+03~ ) 1 = 0= tan n.
2 ] - 1s2+ S
[The denominator 1-s2 + s4 does not vanish unless a2= b2.].'. 0 + 01 + 0 03+04= 2. nr radians
=an even multiple of 7r radians.
EXAMPLES. XLIX.
Prove that
1. cos 4 = cos4 - 6 cos2 sin2 0 + sin40.
2. sin 60= 6 cos5 0 sin 0 - 20 cos3 0 sin3 0 + 6 cos 0 sin5 8.
3. sin 70= 7 cos6 8 sin 0 - 35 cos4 0 sin3 0 + 21 cos2 0 sin5 0 - sin7 0.
4. cos 90 = cos9 - 36 cos7 0 sin2 0 +126 cos5 0 sin4 0
- 84 cos3 8 sin6 0 + 9 cos 0 sin8 0.
5. cos 80 = cos8 0- 28 cos6 0 sin2 0 + 70 cos4 0 sin4 8
- 28cos2 sin6 + sin 0.
Write down, in terms of tan 0, the values of
6. tan 50. 7. tan 70. 8. tan 90.
9. Prove that the last terms in the expressions for cos 110 and
sin 110 are
- 11 cos 0 sin10 0 and - sinl 0.
10. Prove that the last terms in the expressions for sin 80 and sin 90
are - 8 cos 0 sin7 0 and sin9 0 respectively.
11. When n is odd, prove that the last terms in the expansions of
sin nO and cos nO are respectively
n-1 n-1
(- 1) 2 sinn" and n( -1) 2 coS sinn-l0.
12. When n is even, prove that the last terms in the expansion of
sin nO and cos n0 are respectively
n-2 n
n ( -1) 2 cos 0 sinn-l 0 and ( -1) 2sin 0.
13. If a, /, and y be the roots of the equation
X3 +pX2 + qx +p = 0,
prove that tan-l a +tan-1 3 + tan- y = n-r radians,
except in one particular case.