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in Trigonometry by (10.9k points)
Prove that

Cot18+Cot30=Cosec12

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1 Answer

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by (10.9k points)
edited by

LHS = Cot18+Cot30

= \({cos18^o\over sin18^o}+ {\sqrt3}\)

=\({cos18^o+\sqrt3sin18^o \over sin18^o}\)

=\({2(sin30^ocos18^o+\cos30^osin18^o )\over sin18^o}\)

=\({2sin(30^o+18^o)\over sin18^o}\)

=\({2sin48^o\over sin18^o}\)

=\({2sin48^osin12^o\over sin18^osin12^o}\)

=\({cos36^o -cos60^o\over sin18^osin12^o}\)

=\({{{\sqrt{5} +1\over 4}-{1\over2} }\over sin18^osin12^o}\)

=\({{{\sqrt{5} -1\over 4}}\over sin18^osin12^o}\)

= \({{sin18^o}\over sin18^osin12^o}\)

= \({cosec12^o}\)

=RHS proved

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