Advanced book on Mathematics Olympiad

244 4 Geometry and Trigonometry

Example.Prove that

∏∞

n= 1

1

1 −tan^22 −n

=tan 1.

Solution.The solution is based on the identity

tan 2x=

2 tanx

1 −tan^2 x

.

Using it we can write

∏N

n= 1

1

1 −tan^22 −n

=

∏N

n= 1

tan 2−n+^1

2 tan 2−n

=

2 −N

tan 2−N

tan 1.

Since limx→ 0 tanxx=1, when lettingN→∞this becomes tan 1, as desired. 

693.In a circle of radius 1 a square is inscribed. A circle is inscribed in the square and
then a regular octagon in the circle. The procedure continues, doubling each time
the number of sides of the polygon. Find the limit of the lengths of the radii of the
circles.

694.Prove that
(
1 −

cos 61◦

cos 1◦

)(

1 −

cos 62◦

cos 2◦

)

···

(

1 −

cos 119◦

cos 59◦

)

= 1.

695.Evaluate the product

( 1 −cot 1◦)( 1 −cot 2◦)···( 1 −cot 44◦).

696.Compute the product

(

√

3 +tan 1◦)(

√

3 +tan 2◦)···(

√

3 +tan 29◦).

697.Prove the identities

(a)

(

1

2

−cos

π

7

)(

1

2

−cos

3 π

7

)(

1

2

−cos

9 π

7

)

=−

1

8

,

(b)

(

1

2

+cos

π

20

)(

1

2

+cos

3 π

20

)(

1

2

+cos

9 π

20

)(

1

2

+cos

27 π

20

)

=

1

16

.

698.Prove the identities

(a)

∏^24

n= 1

sec( 2 n)◦=− 224 tan 2◦,

(b)

∏^25

n= 2

(2 cos( 2 n)◦−sec( 2 n)◦)=−1.