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LHS

cot A - cot 2A = (cosA/sinA)- (cos2A/sin2A)

=(cosA sin2A -sinA cos2A)/(sinA sin2A)

**∵ cos2A =cos²A - sin²A**

**and, sin2A= 2sinA cosA**

putting the values,

(2sinA cos²A -sinA (cos²A -sin²A))/(2sin²A cosA)

taking common sinA and cancelling sinA ,

(cos²A + sin²A)/(2sinA cosA)

=1/sin2A

= cosec2A = RHS (proved)

cot A - cot 2A = (cosA/sinA)- (cos2A/sin2A)

=(cosA sin2A -sinA cos2A)/(sinA sin2A)

putting the values,

(2sinA cos²A -sinA (cos²A -sin²A))/(2sin²A cosA)

taking common sinA and cancelling sinA ,

(cos²A + sin²A)/(2sinA cosA)

=1/sin2A

= cosec2A = RHS (proved)

There is also another fourmula

cot A + cot 2A = 2 tan 2A

Proof :

cot A - cot 2A = 1/tan A - 1/tan 2A

tan 2A = 2tan A / 1 - tan^2 A

So

1/tan A - (1 - tan^2 A ) / 2tan A

= 1/tan A ( 2 - 1 + tan^2 A ) = 1 + tan^2 A / tan A

= 2 * 1/sin 2A = 2 cosec 2A

since sin 2A = 2tan A / 1 + tan^2 A