cot x. cot 2x - cot 2x. cot3x - cot x. cot3x = 1

I am unable to get the LHS equal to RHS

Hi CHINMAYI,

L.H.S=cot x. cot 2x - cot 2x. cot3x - cot x. cot3x

=cot x. cot 2x - cot3x(cot 2x +cot x)

=cotx .cot 2x - cot (x+2x) .(cot 2x+ cot x)                       [we can write cot3x as cot(x+2x)]

=cotx.cot2x- {(cot x. cot 2x -1)/(cot x + cot 2x)}.(cot x + cot 2x)            [since cot(a+b)=(cot a. cot b -1)/(cot b+cot a)]

=cot x. cot 2x -cot x. cot 2x +1

=1

=R.H.S

cot (3x) = cot ( 2x+x )Apply, cot (A+B) = (cot A.cot B -1)/ cot A + cot BSo,cot (3x) = ( cot 2x.cot x - 1 ) / cot 2x + cot xNow, cross multiplycot 3x ( cot 2x + cot x ) = cot 2x.cot x - 1cot 2x.cot 3x + cot x.cot 3x = cot 2x.cot x - 1cot 2x.cot x - cot 2x.cot 3x - cot x.cot 3x = 1