Question

Prove that cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1

Answer 1

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

Answer 2

Another method: 

Find out, what's the biggest angle here? It's "cot 3x"

cot (3x) = cot ( 2x+x )

Apply, cot (A+B) = (cot A.cot B -1)/ cot A + cot B

So,

cot (3x) = ( cot 2x.cot x - 1 ) / cot 2x + cot x

Now, cross multiply

cot 3x ( cot 2x + cot x ) = cot 2x.cot x - 1

cot 2x.cot 3x + cot x.cot 3x = cot 2x.cot x - 1

cot 2x.cot x - cot 2x.cot 3x - cot x.cot 3x = 1