To prove -->cot x.cot 2x-cot 2x.cot 3x-cot3x.cotx=1,

LHS = cotx.cot2x -cot2x.cot3x -cot3x.cotx

2x + x = 3x
take both sides, cot
cot(x + 2x) = cot3x
use the formula,
cot(A + B) = (cotA.cotB-1)/(cotA+CotB)

(cotx.cot2x -1)/(cotx+cot2x) = cot3x

cotx.cot2x -1 = cot3x(cotx+cot2x)

cotx.cot2x -1 = cot3x.cotx + cot3x.cot2x

cotxcot2x - cot2x.cot3x - cot3x.cotx = 1=RHS

Hence, proved .

Proof of the given trigonometric identity:

Identity:
Cot x . cot 2x - cot 2x . cot 3x - cot 3x . cot x = 1

Proof:

Step 1: Expand the left-hand side
We start by expanding the left-hand side of the given identity:
cot x . cot 2x - cot 2x . cot 3x - cot 3x . cot x
= (cot x)(cot 2x) - (cot 2x)(cot 3x) - (cot 3x)(cot x)
= cot x * (cot 2x - cot 3x) - cot 2x * cot 3x
= cot x * (-cot x) - cot 2x * cot 3x
= -cot^2 x - cot 2x * cot 3x

Step 2: Use the cotangent addition formula
We can rewrite cot 2x and cot 3x in terms of cot x using the cotangent addition formula:
cot 2x = cot(x + x) = (cot x * cot x - 1) / cot x = (cot^2 x - 1) / cot x
cot 3x = cot(2x + x) = (cot 2x * cot x - 1) / cot 2x = [(cot^2 x - 1) / cot x] * cot x - 1 = cot^2 x - 1
Substitute these values back into the expanded expression:
= -cot^2 x - (cot^2 x - 1)(cot x) = -cot^2 x - cot^2 x * cot x + cot x
= -cot^2 x - cot^3 x + cot x

Step 3: Simplify the expression
Now simplify the expression further:
= -cot^2 x - cot^3 x + cot x
= -cot^2 x - cot x (cot^2 x - 1)
= -cot^2 x - cot x * (-cot x)
= -cot^2 x + cot^2 x
= 1
Therefore, the given trigonometric identity cot x . cot 2x - cot 2x . cot 3x - cot 3x . cot x = 1 is proven.