Question 295301
Please help me prove this Trig identity:
{{{ (tany+coty)sinycosy = 1 }}}
define opp = opposite side, adj = adjacent side, hyp = hypotenuse
and hypotenuse is side opposite to the right angle in a right triangle
and sohcahtoa --> sin = opp/hyp, cos = adj/hyp, tan = opp/adj
and sin/cos = opp/hyp * hyp/adj = opp/adj = tan
and cos/sin = adj/hyp * hyp/opp = adj/opp = 1/tan = cot
{{{ (tany + coty)sinycosy = 1 }}}
replacing tan and cot
{{{ (siny/cosy + cosy/siny)sinycosy = 1 }}}
distribute
{{{ siny/cosy * sinycosy + cosy/siny * sinycosy = 1 }}}
{{{ siny * siny + cosy * cosy = 1 }}}
{{{ sin^2(y) + cos^2(y) = 1 }}}
if circle has radius 1:
triangle 30-60-90 with 30 degree angle at center of circle, and hypotenuse being the radius:
sin 30 = opp/hyp = opp/1 = opp
cos 30 = adj/hyp = adj/1 = adj
Pythagorean Theorem --> opp^2 + adj^2 = hyp^2 --> sin^2 + cos^2 = 1
{{{ 1 = 1 }}}
done