654 Geometry and Trigonometry
as desired.
(Gazeta Matematica ̆(Mathematics Gazette, Bucharest), proposed by D. Andrica)
661.We would like to eliminate the square root, and for that reason we recall the trigono-
metric identity
1 −sint
1 +sint=
cos^2 t
( 1 +sint)^2.
The proof of this identity is straightforward if we express the cosine in terms of the sine
and then factor the numerator. Thus if we substitutex =sint, thendx=costdtand
the integral becomes
∫
cos^2 t
1 +sint
dt=∫
1 −sintdt=t+cost+C.Sincet=arcsinx, this is equal to arcsinx+
√
1 −x^2 +C.
(Romanian high school textbook)662.We will prove that a function of the formf (x, y)=cos(ax+by),a, bintegers, can
be written as a polynomial in cosx, cosy, and cos(x+ky)if and only ifbis divisible
byk.
For example, ifb=k, then from
cos(ax+ky)=2 cosxcos((a± 1 )x+ky)−cos((a± 2 )x+ky),we obtain by induction on the absolute value ofathat cos(ax+by)is a polynomial in
cosx,cosy,cos(x+ky). In general, ifb=ck, the identity
cos(ax+cky)=2 cosycos(ax+(c± 1 )ky)−cos(ax+(c± 2 )ky)together with the fact that cosaxis a polynomial in cosx allowsan inductive proof of
the fact that cos(ax+by)can be written as a polynomial in cosx, cosy, and cos(x+ky)
as well.
For the converse, note that by using the product-to-sum formula we can write any
polynomial in cosines as a linear combination of cosines. We will prove a more general
statement, namely that if a linear combination of cosines is a polynomial in cosx, cosy,
and cos(x+ky), then it is of the form
∑m⎡
⎣bmcosmx+∑
0 ≤q<|p|cm,p,q(cos(mx+(pk+q)y)+cos(mx+(pk−q)y))⎤
⎦.
This property is obviously true for polynomials of degree one, since any such poly-
nomial is just a linear combination of the three functions. Also, any polynomial in