Can you please explain why the following identity is true? Where does it come from: pi/24 = cot^(-1)(2+sqrt(2)+sqrt(3)+sqrt(6)). Thanks for any help.
cot-1( 2 + sqrt(2) + sqrt(3) + sqrt(6) ) = pi/24
cot [ pi/24] = 2 + sqrt(3) + sqrt(2) + sqrt(6)
cot [pi/24] =
cot [(1/2)pi/12]] =
1/ tan [(1/2)pi/12)] =
1/ [sqrt ( 1 - cos(pi/12))/ (1 + cos(pi/12)] =
sqrt( 1 + cos(pi/12)) / sqrt( 1 - cos(pi/12) ) = [multiply top/bottom by sqrt( 1 + cos(pi/12)) ]
[1 + cos (pi/12)] / sqrt( 1 - cos^2(pi/12)) =
[1 + cos(pi/12)] / sqrt( sin^2(pi/12)) =
[1 + cos(pi/12) ] / sin(pi/12) =
1/sin(pi/12) + cos(pi/12) / sin(pi/12) =
csc[pi/12] + cot[pi/12] =
1/ sin [ pi/12] + 1 / [tan [pi/12] =
1 / [ sin[ pi/4 - pi/6]] + 1 / [ tan[ pi/4 - pi/6] ]
1/ [ sin(pi/4)cos(pi/6) - sin(pi/6)cos(pi/4)] + 1/ ( [tan(pi/4] - tan[pi/6]] / [1 + tan(pi/4)tan(pi/6)] ) =
1 / [ sqrt(2)sqrt(3)/4] - (sqrt(2)/4) ] + [ 1 + 1/sqrt(3)]/ [1 - 1/sqrt(3)] =
4 / [ sqrt(6) - sqrt(2)] + [ (sqrt(3) + 1)/ sqrt(3))] / ( [ sqrt(3) - 1]/ sqrt(3) )
4 [ sqrt(6) + sqrt(2)]/ 4 + [ sqrt(3) + 1] / [sqrt(3) - 1] = [ multiply the second term by [sqrt(3) + 1] on top/bottom ]
sqrt(6) + sqrt(2) + [sqrt(3) + 1]^2 / 2 =
sqrt(6) + sqrt(2) + [3 + 2sqrt(3) + 1] / 2 =
sqrt(6) + sqrt(2) + [ 4 + 2 sqrt(3)] / 2 =
sqrt(6) + sqrt(2) + 2 + sqrt(3) =
2 + sqrt(3) + sqrt(2) + sqrt(6) ..........and the left side = the right side
cot-1( 2 + sqrt(2) + sqrt(3) + sqrt(6) ) = pi/24
cot [ pi/24] = 2 + sqrt(3) + sqrt(2) + sqrt(6)
cot [pi/24] =
cot [(1/2)pi/12]] =
1/ tan [(1/2)pi/12)] =
1/ [sqrt ( 1 - cos(pi/12))/ (1 + cos(pi/12)] =
sqrt( 1 + cos(pi/12)) / sqrt( 1 - cos(pi/12) ) = [multiply top/bottom by sqrt( 1 + cos(pi/12)) ]
[1 + cos (pi/12)] / sqrt( 1 - cos^2(pi/12)) =
[1 + cos(pi/12)] / sqrt( sin^2(pi/12)) =
[1 + cos(pi/12) ] / sin(pi/12) =
1/sin(pi/12) + cos(pi/12) / sin(pi/12) =
csc[pi/12] + cot[pi/12] =
1/ sin [ pi/12] + 1 / [tan [pi/12] =
1 / [ sin[ pi/4 - pi/6]] + 1 / [ tan[ pi/4 - pi/6] ]
1/ [ sin(pi/4)cos(pi/6) - sin(pi/6)cos(pi/4)] + 1/ ( [tan(pi/4] - tan[pi/6]] / [1 + tan(pi/4)tan(pi/6)] ) =
1 / [ sqrt(2)sqrt(3)/4] - (sqrt(2)/4) ] + [ 1 + 1/sqrt(3)]/ [1 - 1/sqrt(3)] =
4 / [ sqrt(6) - sqrt(2)] + [ (sqrt(3) + 1)/ sqrt(3))] / ( [ sqrt(3) - 1]/ sqrt(3) )
4 [ sqrt(6) + sqrt(2)]/ 4 + [ sqrt(3) + 1] / [sqrt(3) - 1] = [ multiply the second term by [sqrt(3) + 1] on top/bottom ]
sqrt(6) + sqrt(2) + [sqrt(3) + 1]^2 / 2 =
sqrt(6) + sqrt(2) + [3 + 2sqrt(3) + 1] / 2 =
sqrt(6) + sqrt(2) + [ 4 + 2 sqrt(3)] / 2 =
sqrt(6) + sqrt(2) + 2 + sqrt(3) =
2 + sqrt(3) + sqrt(2) + sqrt(6) ..........and the left side = the right side