Plane trigonometry, by S.L. Loney.

[Exs. XLVI.] LOGARITHMS TO BASE 10. 305
19. If tan2 < 1, prove that
1 1
tan2 - tan4 0+ tans6 -... ad inf.
2 3 -
1 1
= sin2 0 + sin4 0 + ~ sin6 0 +... ad inf.
2 3
20. Prove that, if 20 be not a multiple of r-,
1 1
log cot = cos 20 + cos3 20 + cos5 20 +... ad inf.
3 5
21. Prove that the coefficient of x" in the expansion of
{log (1 +x)}2
~~is 2(-1) [ 1+ +.
22. Use the methods of Arts. 259 and 260 to prove that
log1o2= 30103...
and loglo3= 47712....
23. Draw the curve y=logex.
[If x be negative, y is imaginary; when x is zero, y equals - o; when
x is unity, y is nothing; when x is positive and > 1, y is always positive;
when x is infinity, y is infinity also.]
24. Draw the curve y=logo x and state the geometrical relation
between it and the curve of the last example.
[Use Art. 147.]
25. Draw the curve y = a.
261. The two following limits will be required in the
next chapter but one.
262. To prove that the value of cos -, when n is
n/
infinite, is unity.
We have cos a= (1-sin2).
n \ n1
F\ nt) n n sin2
L. T. 20