F.4 AM

Question a
tanA + tanB
= sum of roots
= -p
tanA tanB
= product of roots
= q
Question b
tan(A + B)
= (tanA + tanB) / (1 - tanA tanB)
= -p/(1 - q)
= p / (q - 1)
Question c
tan2(A + B)
= 2tan(A + B)/ [1 - tan2 (A + B)]
= 2p/(q - 1) / [1 - p2 /(q - 1)2 ]
=2p(q - 1)2 / [(q - 1)2 - p2 ]
= 2p(q - 1)2 /(p + q - 1)(q - p - 1)
Question d
sec2 (A + B)
= 1 + tan2 (A + B)
= 1 + p2 /(q - 1)2
= [(q - 1)2 + p2 ]/(q - 1)2
cos2 (A + B)
= 1 / sec2 (A + B)
= (q - 1)2 / [(q - 1)2 + p2 ]