242 4 Geometry and Trigonometry
allows us to prove inductively that 2 coshkt 20 is an integer once we show that 2 cosht 20
is an integer. It would then follow that 2 coshFnt 20 is an integer as well. And indeed
2 cosht 20 =
√
2 + 2 a 1 =14. This completes the proof of part (a).
To prove (b), we obtain in the same manner2 +
√
2 + 2 an=(
2 coshFnt 0
4) 2
,
and again we have to prove that 2 cosh√ t 40 is an integer. We compute 2 cosht 40 =
2 +√
2 + 2 an=√
2 + 14 =4. The conclusion follows. 686.Compute the integral
∫
dx
x+
√
x^2 − 1687.Letn>1 be an integer. Prove that there is no irrational numberasuch that the
number
n
√
a+√
a^2 − 1 +n√
a−√
a^2 − 1is rational.4.2.4 Telescopic Sums and Products in Trigonometry...............
The philosophy of telescopic sums and products in trigonometry is the same as in the
general case, just that here we have more identities at hand. Let us take a look at a slightly
modified version of an identity of C.A. Laisant.
Example.Prove that
∑nk= 0(
−
1
3
)k
cos^3(
3 k−nπ)
=
3
4
[(
−
1
3
)n+ 1
+cos
π
3 n]
.
Solution.From the identity cos 3x=4 cos^3 x−3 cosx, we obtain
cos^3 x=1
4
(cos 3x+3 cosx).Then
∑nk= 0(
−
1
3
)k
cos^3(
3 ka)
=
1
4
∑nk= 0[(
−
1
3
)k
cos(
3 k+^1 a)
−
(
−
1
3
)k− 1
cos(
3 ka