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Question
Prove the following:
sin47° + sin61° − sin11° − sin25° = cos7°
Solution
L.H.S. = sin47° + sin61° − sin11° − sin25°
= (sin47° − sin25°) + (sin61° − sin11°)
= `2cos((47^circ + 25^circ)/2)*sin ((47^circ - 25^circ)/2) + 2cos((61^circ + 11^circ)/2)*sin((61^circ - 11^circ)/2)`
= 2cos36° · sin11° + 2cos36° · sin25°
= 2cos36° (sin25° + sin11°)
= `2cos36^circ xx 2sin((25^circ + 11^circ)/2)*cos((25^circ - 11^circ)/2)`
= 4cos36° · sin18° · cos7°
= `4 xx (sqrt(5) + 1)/4 xx (sqrt(5) - 1)/4 xx cos7^circ ...[because cos36^circ = (sqrt(5) + 1)/4, sin 18^circ = (sqrt(5) - 1)/4]`
= `(4(5 - 1))/16 cos7^circ`
= cos7°
= R.H.S.
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