Show that cos - cos /sin - sin = a/b, Mathematics

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A ladder sets against a wall at an angle α to the horizontal.  If the foot is pulled away from the wall through a distance of 'a', so that is slides a distance 'b' down the wall making an angle β with the horizontal. Show that cosα - cos β/sin β - sin α = a/b

Ans:    Let CB = x m. Length of ladder remains same

Cos α = CB/CA     ∴ ED = AC  Let Ed be

Cos α = x/h       ∴ ED = AC = h

x = hcos α                .........(1)

cos β =DC CB/ED

cos β = a + x/h

a + x = hcos β

x = hcos β - a              .........(2)

from (1) & (2)

hcos α = hcos β - a

hcos α - hcos β = - a

-a = h(cosα - cosβ)      .........(3)

Sin α = Sin α = AE + EB/AC

Sin α =b + EB/ h

hSin α - b = EB

EB = hSin α - b          .........(4)

Sin β = EB /DE

Sin β = EB/h

EB = hSin β             .........(5)

From (4) & (5)

hSin β = hSin α - b

b = hsin α - hSin β

-b = h(Sin β - Sin α)   .........(6)

Divide equation (3) with equation (6)

- a/-b = h(cosα - cos β )/ h(sin β - Sinα )

a/b = Cosα - Cosβ/Sinβ - Sinα


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