If A and B are the roots of 3x^2-10x-25=0, then the value of 3^2(A+B)-10(A+B)(A+B)-25^2(A+…

Question

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

Accepted Answer

$\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}=\dfrac{10/3}{1+25/3}=\dfrac5{14}$.

$$3\sin^2\phi-10\sin\phi\cos\phi-25\cos^2\phi=\cos^2\phi[3\tan^2\phi-10\tan\phi-25],\quad \phi=A+B.$$

Substituting $\tan\phi=\dfrac5{14}$ and $\cos^2\phi=\dfrac{196}{221}$ gives the value $-25.$

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