Geometry and Trigonometry 671Using the identity sina−cosa=√
2 sin(a− 45 ◦)in the numerators, we transform this
further into
√
2 sin( 1 ◦− 45 ◦)·√
2 sin( 2 ◦− 45 ◦)···√
2 sin( 44 ◦− 45 ◦)
sin 1◦sin 2◦···sin 44◦=(
√
2 )^44 (− 1 )^44 sin 44◦sin 43◦···sin 1◦
sin 44◦sin 43◦···sin 1◦.
After cancellations, we obtain 2^22.
696.We can write
√
3 +tann◦=tan 60◦+tann◦=sin 60◦
cos 60◦+
sinn◦
cosn◦
=
sin( 60 ◦+n◦)
cos 60◦cosn◦= 2 ·
sin( 60 ◦+n◦)
cosn◦= 2 ·
cos( 30 ◦−n◦)
cosn◦.
And the product telescopes as follows:
∏^29n= 1(
√
3 +tann◦)= 229∏^29
n= 1cos( 30 ◦−n◦)
cosn◦= 229 ·
cos 29◦cos 28◦···cos 1◦
cos 1◦cos 2◦···cos 29◦= 229.
(T. Andreescu)
697.(a) Note that1 −2 cos 2x= 1 − 2 (2 cos^2 x− 1 )= 3 −4 cos^2 x=−
cos 3x
cosx.
The product becomes
(
−1
2
) 3
cos^37 π
cosπ 7·
cos^97 π
cos^37 π·
cos^277 π
cos^97 π=−
1
8
·
cos^277 π
cosπ 7.
Taking into account that cos^277 π=cos( 2 π−π 7 )=cosπ 7 , we obtain the desired identity.
(b) Analogously,1 +2 cos 2x= 1 + 2 ( 1 −2 sin^2 x)= 3 −4 sin^2 x=sin 3x
sinx,
and the product becomes1
24sin^320 π
sin 20 π·
sin^920 π
sin^330 π·
sin^2720 π
sin^920 π·
sin^8120 π
sin^2720 π=
1
16
sin^8120 π
sin 20 π