Advanced book on Mathematics Olympiad

4.2 Trigonometry 243

This telescopes to

1

4

[(

−

1

3

)n

cos

(

3 n+^1 a

)

−

(

−

1

3

)− 1

cosa

]

.

Fora= 3 −nπ, we obtain the identity from the statement. 

Test your skills against the following problems.

688.Prove that

27 sin^39 ◦+9 sin^327 ◦+3 sin^381 ◦+sin^3243 ◦=20 sin 9◦.

689.Prove that

1

cot 9◦−3 tan 9◦

+

3

cot 27◦−3 tan 27◦

+

9

cot 81◦−3 tan 81◦

+

27

cot 243◦−3 tan 243◦

=10 tan 9◦.

690.Prove that

1

sin 45◦sin 46◦

+

1

sin 47◦sin 48◦

+···+

1

sin 133◦sin 134◦

=

1

sin 1◦

.

691.Obtain explicit values for the following series:

(a)

∑∞

n= 1

arctan

2

n^2

,

(b)

∑∞

n= 1

arctan

8 n

n^4 − 2 n^2 + 5

.

692.Forn≥0 let

un=arcsin

√

n+ 1 −

√

n

√

n+ 2

√

n+ 1

.

Prove that the series

S=u 0 +u 1 +u 2 +···+un+···

is convergent and find its limit.

Now we turn to telescopic products.