Plane trigonometry, by S.L. Loney.

416 TRIGONOMETRY. [Exs. LXII.]
I 1
11. sin 20 cos2 8 - sin 40 cos2 20 + 4 sin 80 cos2 40 -...... to n terms.
2 4
1 1
12. sin 28 sin2 0 + sin 40 sin2 20 + I sin 80 sin2 40 +...... to n terms.
sin 0 sin 20 sin 30
13. - +-......to n terms.
cos 0 + cos 2 + cos 0 + cos 40 cos 0 + cos 60
14. tan2 a tan 2a + 1 tan2 2a tan 4a + 1 tan2 4a tan 8a +...... ad inf.
2 22
1 1 1 1
15. cos3 0 - cos3 33+ cos3 32 - cos3 33 +...... to n terms.
16. sins3 + 3 sin3s - +32 sin3s +...... to n terms.
3 3 3 2
17. + + Co 32....
cot 0 - 3 tan + cot 30 - 3 tan 30 + cot 32 - 3 tan 320 +
to n terms.
cos - cos 30 cos 3 - cos 3'0 cos 320 - cos 338
18.i -- + 32.......
sin 30 +3 sin 320 3 sin 30.
to n terms.
4 6 8
19. tan-' 4 + tan-l + tan-' +...... to n terms.
1+3.4 1+8.9 1+15.16
1 1 1 1
20. tan-' + tan-' - + tan + n- +....... to n terms.
3 7 i-3 21
1 2 2-l1
21. tan- - + tan-' +...... +tan-l1 2-.... ad inf.
nn *9 1 + 22-1-1
22. si 2 6 sin -.....
+ sin-1 dn - 1n_-Z-f
- sin-_1 + --...... ad inf.
J/ (n + 1)
Expansions.
357. In some branches of higher Mathematics it is
desirable to be able to expand certain quantities in a
series of ascending powers.
As an example we will expand
log (1 - 2a cos 0 + a2)
in ascending powers of a.