start from RHS as it is more complex
1/ (sin 2pi/ 7) + 1/(sin 3pi/7)
= 1/ ( sin 5pi/7) + 1/( sin 3pi/7) as sin 2pi/7 = sin (pi-2pi/7) = sin 5pi/7 and we can add with out a faction of the form pi/14)
= ( sin 3pi/7 + sin 5pi/7) / ( sin 5pi/7 sin 3pi/7)
= (2 sin 4pi/7 cos pi/7)/ ( ( sin 5pi/7 sin 3pi/7)
as sin 4pi/7 = sin 3pi/7 and one is in numerator and another in denominator
= (2 sin 3pi/7 cos pi/7)/( sin 5pi/7 sin 3pi/7)
= (2 cos pi/7)/ ( sin 5pi/7)
= (2 cos pi/7)/( sin 2pi/7) as sin 5pi/7 = sin 2pi/7
= ( 2 cospi/7 )/ ( 2 sin pi/7 cos pi/7)
= 1/ sin pi/7 = LHS
1 comment:
Nice algorithm!
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