Question

If tan A and tan B are the roots of x² – ax + b = 0, then the value of sin²(A + B) is ______.

Options

• `a^2/(a^2 + (1 - b)^2`

• `a^2/(a^2 + b^2)`

• `a^2/(a^2 + b)^2`

• `b^2/(a^2 + (a - b)^2`

Answer 1

If tan A and tan B are the roots of x² – ax + b = 0, then the value of sin²(A + B) is `underlinebb(a^2/(a^2 + (1 - b)^2`.

Explanation:

tan A + tan B = a and tan A tan B = b

tan (A + B) = `(tan A + tan B)/(1 - tan A tan B) = a/(1 - b)`

Now, sin² (A + B) = `1/2 [1 - cos 2(A + B)]`

= `1/2[1 - (1 - tan^2 (A + B))/(1 + tan^2(A + B))]`

= `[(tan^2(A + B))/(1 + tan^2 (A + B))]`

= `(a^2/(1 - b)^2)/([(a^2 + (1 - b)^2)/(1 - b)^2]`

= `a^2/(a^2 + (1 - b)^2`