If tan A and tan B are the roots of the quadr | vedclass

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(NONE) Given the quadratic equation $3x^2 - 10x - 25 = 0$ with roots $	an A$ and $	an B$.
From the properties of roots,$	an A + 	an B = \frac{10}

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(NONE) Given the quadratic equation $3x^2 - 10x - 25 = 0$ with roots $	an A$ and $	an B$.
From the properties of roots,$	an A + 	an B = \frac{10}{3}$ and $	an A 	an B = -\frac{25}{3}$.
Using the formula $	an (A + B) = \frac{	an A + 	an B}{1 - 	an A 	an B}$,we get:
$	an (A + B) = \frac{10/3}{1 - (-25/3)} = \frac{10/3}{28/3} = \frac{10}{28} = \frac{5}{14}$.
Let $S = 3 \sin^2 (A + B) - 10 \sin (A + B) \cos (A + B) - 25 \cos^2 (A + B)$.
Divide the entire expression by $\cos^2 (A + B)$:
$S = \cos^2 (A + B) [3 	an^2 (A + B) - 10 	an (A + B) - 25]$.
Since $	an (A + B) = 5/14$,the term in the bracket is $3(5/14)^2 - 10(5/14) - 25 = 3(25/196) - 50/14 - 25 = 75/196 - 700/196 - 4900/196 = -5525/196$.
Alternatively,note that the expression is of the form $f(	an(A+B)) \cdot \cos^2(A+B)$ where $f(x) = 3x^2 - 10x - 25$. Since $	an(A+B) = 5/14$ is a root of $3x^2 - 10x - 25 = 0$,the value of the expression is $0$.

If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3x^2 - 10x - 25 = 0$,then the value of $3 \sin^2 (A + B) - 10 \sin (A + B) \cos (A + B) - 25 \cos^2 (A + B)$ is

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