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

Trigonometry is a branch of mathematics that deals with the relation between the angles and sides of a triangle.

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

We will use the trigonometric formula (cot (A+B) = ((cot A. cot B -1)/ (cot A + cot B))

Explanation:

Consider LHS:

cot x. cot 2x - cot 2x. cot3x - cot x. cot3x

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

cot 3x = cot (2x+x)

Using the formula, 

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

We have,

cot (2x+x) = ((cot 2x. cot x – 1) / (cot 2x + cot x))  

(1) becomes 

cot x. cot 2x- \(\dfrac{cot 2x. cot x – 1}{cot 2x + cot x}\) . (cot 2x+cot x)

= cot x. cot 2x- cot 2x. cot x – 1

= 1 = RHS

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

Hence, proved cot x. cot 2x - cot 2x. cot3x - cot x. cot3x = 1.