L.H.S[i'm using  @ as theta]

 

cos^2 @ /sin^2@ ={ cos(@ - A) cos(@- B) }/ {SIN(@-A) SIN (@-B)}

 

APPLYING DIVIDENDO & COMP.

 

 {COS^2@ - SIN^2@} /COS^2 @+ SIN^2 @ ={ COS( @-A + @ -B )} / {COS (@ - A - @ + B}

 

USING  COS (A+OR - B)

 

 COS 2@ = [COS{ 2@ - ( A+B)}] /COS(A- B)

[ SINCE COS (-@) = COS @]

OR, COS 2@ COS( A - B) = COS 2@ COS (A+B) + SIN 2@ SIN ( A+B)

OR, COS 2@{ COS( A-B) - COS (A+B)} = SIN 2@ SIN ( A+B)

 

SO,

COS2@ / SIN 2@ = {SIN A COS B + COS A SIN B}/ 2 SIN A SIN B

OR, COT 22 = 1/2 [ COT B + COT A]

 

PROVED

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