How will you prove the below identiy ?

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coordinate geometry

Coordinate geometry

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Using inscribed angle theorem we get:-

2B = sum of measure of arcs (AE+ED+DC)
2E = sum of measure of arcs (AB+BC+CD)
2C = sum of measure of arcs(BA+AE+ED)

since sum of measure of all arcs in a circle is 360 degrees.
=> (B+E) = (2CD+AB+BC+AE+ED) = 0.5(360+ CD)
=> (C+E) = (2AB+AE+ED+BC+CD) = 0.5(360 + AB)

Taking sin of both sides
sin(B+E) = sin (180+CD/2) = -sin(CD/2)
sin(C+E) = sin (180+AB/2) = -sin(AB/2)

sin(B+E)         -sin(CD/2)
=> ________    = ___________ ——————————-(1)
sin(C+E)         -sin(AB/2)
By applying cosine rule on the constructed triangles OCD an OAB, we get:-

a^2 = 2*r*r – 2*r*r cos (measure of arc CD) = 2*r*r(1 – cos (measure arc CD)) ———— (2)
d^2 = 2*r*r – 2*r*r cos (measure of arc AB) = 2*r*r(1 – cos (measure arc AB)) —————-(3)

Dividing equation (2) by (3) and taking square root of both sides (considering negative value of the square root only to substitute in equation 1)

a           sqrt(1 – cos CD)
__ = ___________________
d           sqrt(1 – cos AB)

sqrt(1 – (1 – 2*sin^2 CD/2))
=  _____________________________________
sqrt(1 – (1 – 2*sin^2 CD/2))

sqrt(2) * -sin(CD/2)
=   ______________________  ——————————-(4)
sqrt(2) * -sin(AB/2)

FROM EQUATIONS 1 AND 4 we get:-
sin(B+E)          a
________ = ______
sin(C+E)          d

Edited answer

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