32. tan (45°+A) – tan (45° – A)=2 tan 24.

33. tan (45°+A)+tan (45° – A) = 2 sec 24.

125. Functions of 3A.

sin 3A =sin (24+4)=sin 24 cos A+ cos 24 sin A =2 sin A cos2 A+ (1 - 2 sin2 A) sin A

=2 sin A (1 − sin2 A)+(1 − 2 sin2 A) sin A ;

=3 sin A - 4 sin3 A.

Similarly it may be proved that

These formulæ are perfectly general and may be applied to cases of any two angles, one of which is three times the other; thus

cos 6a=4 cos3 2a-3 cos 2a;

sin 943 sin 34 - 4 sin3 3A.

126. To find the value of sin 18°.

Let A=18°, then 5A-90°, so that 24-90° - 34.

.. sin 24 sin (90°-34)= cos 34;

=

... 2 sin A cos A=4 cos3 A-3 cos A..

Divide by cos A (which is not equal to zero);

.. 2 sin A=4 cos2 A-3=4(1 − sin2 A) −3;

... 4 sin2 A +2 sin A-1=0;

Since 18° is an acute angle, we take the positive sign ;

Since 54° and 36° are complementary, sin 54°=cos 36°.

127. The following examples further illustrate the formulæ proved in this chapter.

Example 1. Shew that cos6 a+ sin6 a=1

3

sin22a.

4

The first side=(cos2a+sin2 a) (cos1a + sin1a – cos2 a sin2 a)

=(cos2a+sin2a)2-3 cos2 a sin2a

and since cos 24-cos24 - sin2 A = (cos A + sin A) (cos A – sin ▲), this suggests that we should multiply the numerator and denominator of the left side by cos A - sin A; thus

NOTE. This example has been given to emphasize the fact that in identities involving the functions of 24 and 34 it is sometimes best not to substitute their equivalents in terms of functions of A.

16. (2 cos A+1) (2 cos A − 1)=2 cos 24 +1.

cos A

1+cos 24'1+cos A

19. 4 sin3 a cos 3a + 4 cos3 a sin 3a=3 sin 4a.

[Put 4 sin3a=3 sin a- sin 3a and 4 cos3a=3 cos a + cos 3a.]

20. cos3 a cos 3a+sin3 a sin 3a=cos3 2a.

21. 4 (cos3 20° + cos3 40°)=3(cos 20°+cos 40°).

22. 4 (cos3 10°+sin3 20°)=3 (cos 10° + sin 20°).

23. tan 34-tan 24 - tan Atan 34 tan 24 tan A.