4.2 Trigonometry 237f(x)+f(
x+2 π
3)
+f(
x+4 π
3)
= 3 a.Ifa<0, thens=twould work. Ifa=0, then for somexone of the terms of the above
sum is negative. This is becausef(x)is not identically zero, since its Fourier series is
not trivial. Ifa>0, substitutingx=tin the identity deduced above and using the fact
thatf(t)= 4 a, we obtain
f(
t+2 π
3)
+f(
t+4 π
3)
=−a< 0.Hence eitherf(t+^23 π)orf(t+^43 π)is negative. The problem is solved.
666.Prove the identity
(
1 +itant
1 −itant
)n
=1 +itannt
1 −itannt,n≥ 1.667.Prove the identity
1 −
(
n
2)
+
(
n
4)
−
(
n
6)
+··· = 2 n/^2 cos
nπ
4,n≥ 1.668.Compute the sum
(
n
1
)
cosx+(
n
2)
cos 2x+···+(
n
n)
cosnx.669.Find the Taylor series expansion at 0 of the function
f(x)=excosθcos(xsinθ),whereθis a parameter.670.Letz 1 ,z 2 ,z 3 be complex numbers of the same absolute value, none of which is real
and all distinct. Prove that ifz 1 +z 2 z 3 ,z 2 +z 3 z 1 andz 3 +z 1 z 2 are all real, then
z 1 z 2 z 3 =1.
671.Letnbe an odd positive integer and letθbe a real number such thatπθis irrational.
Setak=tan(θ+kπn),k= 1 , 2 ,...,n. Prove that
a 1 +a 2 +···+an
a 1 a 2 ···an
is an integer and determine its value.672.Find(cosα)(cos 2α)(cos 3α)···(cos 999α)withα= 19992 π.