Question

Prove that cosπ/13cos2π/13cos3π/13cos4π/13cos5π/13cos6π/13=1/64​

Answer 1

LHS=

Cosπ/13 Cos 2π/13 Cos3π /13 Cos4π/13 Cos 5π /13 Cos 6π/13

We have a formula

2Sinθ Cosθ = Sin 2θ

Applying it here

1

= -------------- (2Sinπ/13Cosπ/13)Cos2π/13

2 Sinπ/13

Cos3π/13 Cos4π/13 Cos6π/13

1

=----------------(2 Sin2π/13Cos2π/13)

4 Sinπ/13

Cos3π/13 Cos4π/13 Cos5π/13Cos6π/13

1

=-----------------(2Sin4π/13Cos4π/13)

8Sinπ/13

Cos3π/13 Cos5π/13Cos 6π/13

1

= -------------- Sin8π/13Cos5π/13Cos6π/13

8Sinπ/13

Cos3π/13

We know that

5π

Sin8π /13 = Sin( π - -------)

13

= Sin 5π/13

Putting it here

1

= --------------Sin5π/13 Cos5π/13 Cos6π/13

8 Sin π/13

Cos 3π/13

1

=--------------2Sin5π/13Cos5π/13 Cos6π/13 16Sin π/13

Cos3π/13

1

=---------------- Sin 10π/13 Cos6π/13

16Sinπ/13

Cos 3π/13

We know that

Sin10π /13 = Sin( π - 3π /13)

= Sin 3π/13

1

=------------------ 2Sin3π/13cos 3π/13

32 Sin π/13

Cos 6π/13

1

= ----------------- 2Sin 6π/13 Cos 6π/13

64 Sin π/13

1

= --------------------- Sin 12π /13

64 Sin π/13

1

= ---------------------- Sin (π - π /13 )

64 Sin π/13

1

= ---------------------- Sin π /13

64 Sin π /13

1

= ----------- =RHS

64