Plane trigonometry, by S.L. Loney.

CHAPTER VIII.

THE TRIGONOMETRICAL RATIOS OF MULTIPLE AND
SUBMULTIPLE ANGLES.
105. To find the trigonometrical ratios of an angle 2A
in terms of those of the angle A.
If in the formulae of Art. 88 we put B = A, we have
sin 2A = sin A cos A + cos A sin A = 2 sin A cos A,
cos 2A = cos A cos A - sin A sin A = cos2 A - sin2 A
=(1 - sin2 A) - sin2A = 1 -2 sin2 A,
and also
= cos2 A -(1 - cos2 A) = 2 cos2 A - 1;
and
tan A + tan A 2 tan A
tan 2A = =
1 - tan A. tan A 1 - tan2 A
Now the formulae of Art. 88 are true for all values of
A and B; hence any formulae derived from them are true
for all values of the angles.
In particular the above formulae are true for all values
of A.