UPSC NDA Math Model Paper 8

Given: Let tan α and tan β are the roots of the equation x2+bx+c=0 with b ≠ 0.
By comparing x2+bx+c=0 with the standard quadratic equation ax2+bx+c=0 we have a = 1, b = b and c = c.
⇒tan‌α+tan‌β=−b and tan‌α×tan‌β=c
⇒tan‌α+tan‌β=

sin‌α

cos‌α

+

sin‌β

cos‌β

=−b
⇒tan‌α+tan‌β=

sin(α+β)

cos‌α×cos‌β

=sin(α+β)‌sec‌α‌sec‌β=−b
⇒sin(α+β)‌sec‌α‌sec‌β=−b