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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
Solution
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.
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