Question

State whether the statement is True or False? Also give justification.

`cos  (2pi)/15 cos  (4pi)/15 cos  (8pi)/15 cos  (16pi)/15 = 1/16`

Options

• True

• False

Answer 1

This statement is True.

Explanation:

L.H.S. `cos  (2pi)/15 cos  (4pi)/15 cos  (8pi)/15 cos  (16pi)/15 = 1/16`

= cos24°.cos48°.cos96°.cos192°

= `1/(16  sin 24^circ) [(2 sin 24^circ cos 24^circ)(2 cos 48^circ)(2 cos 96^circ)(2 cos 192^circ)]`

= `1/(16  sin24^circ) [sin 48^circ . 2 cos48^circ (2 cos 96^circ)(2 cos192^circ)]`

  `1/(16  sin 24^circ) [2 sin 48^circ cos 48^circ (2 cos 96^circ)(2 cos 192^circ)]`

= `1/(16  sin24^circ) [sin 96^circ (2 cos96^circ)(2 cos 192^circ)]`

= `1/(16  sin 24^circ) [2 sin 96^circ . cos 96^circ (2 cos 192^circ)]`

= `1/(16  sin 24^circ) 2 sin 192^circ cos 192^circ`

= `1/(16  sin 24^circ) sin 384^circ`

= `1/(16  sin 24^circ) sin(360^circ + 24^circ)`

= `1/(16  sin 24^circ) xx sin 24^circ`  .......[∵ sin(360° + θ) = sin θ]

= `1/16` R.H.S.