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प्रश्न
Prove the following:
tan6° tan42° tan66° tan78° = 1
उत्तर
L.H.S. = tan6° tan42° tan66° tan78°
= `sin6^circ/cos6^circ * sin42^circ/cos42^circ *sin66^circ/cos66^circ *sin78^circ/cos78^circ`
= `((2sin66^circ sin6^circ)(2sin78^circ sin42^circ))/((2cos66^circ cos6^circ)(2cos78^circ cos42^circ)`
= `(cos(66^circ - 6^circ) - cos(66^circ + 6^circ))/(cos(66^circ + 6^circ) + cos(66^circ - 6^circ)) * (cos(78^circ - 42^circ) - cos(78^circ + 42^circ))/(cos(78^circ + 42^circ) + cos(78^circ - 42^circ))`
= `((cos60^circ - cos72^circ)(cos36^circ - cos120^circ))/((cos60^circ + cos72^circ)(cos36^circ + cos120^circ)`
= `((cos60^circ - sin18^circ)(cos36^circ + sin30^circ))/((cos60^circ + sin18^circ)(cos36^circ - sin30^circ)` ...[∵ cos(90° + θ) = – sin θ]
= `((1/2 - (sqrt(5) - 1)/4)((sqrt(5) + 1)/4 + 1/2))/((1/2 + (sqrt(5) - 1)/4)((sqrt(5) + 1)/4 - 1/2))`
= `(9 - 5)/(5 - 1)`
= 1
= R.H.S.
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