Question

if A,B,C and D are the angles of a cyclic quadrilateral show that
1) tan A/2 =cot C/2
2) sin B+D /2= 1
3) sec B/2×sin D/2 =1

Answer 1

For this question we use the relation that cos( 180 - A) = - cos A

For any qudrilateral the sum of the angles is equal to 360 degree.

So, A+ B+ C +D= 360 degree.

=> A+ B = 360 - ( C + D)

=> ( A + B)/2 = [360 - (C+ D)]/2

=> (A +B)/2 = 180 - (C+ D)/2

Now take the cos of both the sides:

=> cos (A +B)/2 = cos [180 - (C+ D)/2]

=> cos ( A+ B)/2 = - cos ( C+ D)/2

Therefore, bringing cos ( C+D)/2 to the other side

=> cos ( A+ B)/2 + cos ( C+ D)/2 = 0



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