Question

prove that :cot 15/2 degrees = square root of 2 + root of 3 + root of 4 + root of 6

Answer 1

cot 7.5 degree = sqrt 6 + sqrt 3 + sqrt 2 +2

cot(7.5) = cot{(45-30)/2} = (cos45 + cos30)/(sin45-sin30) 
= ( 1/sqrt(2) - sqrt(3)/2 )/( 1/sqrt(2) - 1/2) = ( 2 - sqrt(6) ) / ( 2 - sqrt(2) ) 
= sqrt(6) + sqrt(3) + sqrt(2) + 2 

Answer 2

cot 2Ф =  (cot² Ф - 1 )/ 2 cot Ф 
 let Ф = 15 deg          Cot 2 * 15 = cot 30 = √3
2√3 cot Ф = cot² Ф -1    =>    cot 15  = 2 + √3

Apply the above rule for cot 2Ф again with 2Ф = 15 deg