Question

Prove that height is h(1+tanalpha.tanbeta)

Answer 1

Solution

To prove:h=h(1+tanα.tanβ)/2

Proof: if α and β are angles in a trigonomentric(right angled triangle) then,
α+β+90°=180°
α+β=180°-90°
α+β=90
hence, β=90-α

As we know, tanθ=sinθ/cosθ

RHS=h(1+tanα.tanβ)/2
         =h(1+sinα/cosα x sinβ/cosβ)/2
         =h(1+sinαsinβ/cosαcosβ)/2
         =h(cosαcosβ+sinαsinβ/cosαcosβ)/2
         =h(cosαcos(90°-α)+sinαsin(90°-α)/2cosαcos(90°-α)) 
As, sinθ=cos(90-°θ)
         =h(cosαsinα+sinαcosα/2cosαsinα)      
         =h(2sinαcosα/2sinαcosα)
         =h(1)
         =h.

                                Hence proved
                             Hope it helps