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cotxcoty-cot2xcot3x-cot3xcotx=1
cot(a+b) = (cosa cosb - sina sinb) / (sina cosb + sinb cosa) =cosa cosb / (sina cosb + sinb cosa) - sina sinb / (sina cosb + sinb cosa) = 1/ (sina/cosa + sinb/cosb) - 1/(cosb/sinb + cosa/sina) = 1/ (tana + tanb) - 1/ (cota + cotb) So cot(3x) = 1/(tanx + tan2x) - 1/(cotx + cot2x) Hence cotx.cot2x - cot2x.cot3x - cot3x.cotx =cotx.cot2x - (cot2x+cotx).cot3x =cotx.cot2x - (cot2x+cotx).(1/(tanx + tan2x) - 1/(cotx + cot2x)) =cotx.cot2x - (cotx+cot2x)/(tanx+tan2x) + 1 = cotx.cot2x - (cotx+cot2x)/(1/cotx + 1/cot2x) + 1 = cotx.cot2x - (cotx+cot2x)/(cot2x/cotx.cot2x + cotx/cotx.xot2x)) + 1 = cotx.cot2x - (cotx+cot2x)/((cotx+cot2x)/cotx.cot2x) + 1 = cotx.cot2x - cotx.cot2x + 1 = 1 Please approve !
cot(a+b) = (cosa cosb - sina sinb) / (sina cosb + sinb cosa) =cosa cosb / (sina cosb + sinb cosa) - sina sinb / (sina cosb + sinb cosa) = 1/ (sina/cosa + sinb/cosb) - 1/(cosb/sinb + cosa/sina) = 1/ (tana + tanb) - 1/ (cota + cotb) So cot(3x) = 1/(tanx + tan2x) - 1/(cotx + cot2x) Hence cotx.cot2x - cot2x.cot3x - cot3x.cotx =cotx.cot2x - (cot2x+cotx).cot3x =cotx.cot2x - (cot2x+cotx).(1/(tanx + tan2x) - 1/(cotx + cot2x)) =cotx.cot2x - (cotx+cot2x)/(tanx+tan2x) + 1 = cotx.cot2x - (cotx+cot2x)/(1/cotx + 1/cot2x) + 1 = cotx.cot2x - (cotx+cot2x)/(cot2x/cotx.cot2x + cotx/cotx.xot2x)) + 1 = cotx.cot2x - (cotx+cot2x)/((cotx+cot2x)/cotx.cot2x) + 1 = cotx.cot2x - cotx.cot2x + 1 = 1
Please approve !
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