Q13/062) prove cot6°cot42°cot66° cot78° =1

we know  cot x cot (60+ x) cot (60-x) = cot 3x  ..(1)

put x= 18 to get

cot 18  cot 42 cot 78  = cot 54

or cot 42 cot 78 = cot 54 tan 18 = cot 54 cot 72

so cot6°cot42°cot66° cot78 = cot 6 cot 54 cot 72 cot 66 … (2)

from (1) putting x = 6 we get

cot 6 cot 54 cot 66 = cot 18 or tan 72 …(3)

putting this on RHS of (2) we get

cot6°cot42°cot66° cot78 = (cot 6 cot 54 cot 66) cot 72

= tan 72 cot 72

= 1