Question

X=theta .....If alpha and beta are two distinct roots of the equation atanX+bsecX =c . Prove that tan (alpha+beta)= 2ac/a^2-c^2

Plzz on wed i have exam.....get me the answer soon...its like from conditional identities trignometry..!

Answer 1

It seems , we are late for answering. we are sorry for late.
Well, this question can be solved with help of simple algebraic concepts.
Let's see,

Given, tanα and tanβ are the root of atanx + bsecx = c
first of all you should resolve the equation in the form of quadratic.
atanx + bsecx = c
⇒atanx - c = - bsecx
squaring both sides,
⇒(atanx - c)² = (-bsecx)²
⇒a²tan²x + c² -2actanx = b²sec²x
⇒a²tan²x + c² - 2actanx = b² (1 + tan²x) [ ∵ sec²Ф - tan²Ф = 1]
⇒ (a² - b²)tan²x -2actanx + (c² - b²) = 0
It likes quadratic equation.
question said , tanα and tanβ are the roots of this equation.
so, sum of roots = tanα + tanβ = 2ac/(a² - b²)
product of roots = tanα.tanβ = (c² - b²)/(a² - b²)

∵ tan(α + β) = (tanα + tanβ)/(1 - tanα.tanβ)
= {2ac/(a² - b²)}/{1 - (c² - b²)/(a² - b²)}
= 2ac/(a - c²)

Hence proved//