If $ an \alpha$ and $ an \beta$ are the roots of the equation $x^2+p x+q$ $=0$, then the value of $\sin ^2(\alpha+\beta)+p \cos (\alpha+\beta) \sin (\alpha$ $+\beta)+q \cos ^2(\alpha+\beta)$ is

Question

If $	an \alpha$ and $	an \beta$ are the roots of the equation $x^2+p x+q$ $=0$, then the value of $\sin ^2(\alpha+\beta)+p \cos (\alpha+\beta) \sin (\alpha$ $+\beta)+q \cos ^2(\alpha+\beta)$ is

Aicharya · Accepted Answer

Given that, $	an \alpha$ and $	an \beta$ are the roots of the equation $x^2+p x+q=0$ $ 	herefore \quad 	an \alpha+	an \beta=-p 	ext { and } 	an \alpha \cdot 	an \beta=q $ Now, $\quad 	an (\alpha+\beta)=\frac{	an \alpha+	an \beta}{1-	an \alpha 	an \beta}$ $ \begin{aligned} & =\frac{-p}{1-q}=\frac{p}{q-1} \ & \sec (\alpha+\beta)=\sqrt{1+	an ^2(\alpha+\beta)} \ & \sec (\alpha+\beta)=\sqrt{1+\frac{p^2}{(q-1)^2}} \ & 	herefore \quad \cos (\alpha+\beta)=\frac{1}{\sqrt{1+\frac{p^2}{(q-1)^2}}} \ & \sin ^2(\alpha+\beta)+p \cos (\alpha+\beta) \sin (\alpha+\beta) \ & =\cos ^2(\alpha+\beta)\left[	an ^2(\alpha+\beta)+p 	an (\alpha+\beta)+qight] \ & =\frac{1}{1+\frac{p^2}{(q-1)^2}}\left[\frac{p^2}{(q-1)^2+\frac{p^2}{q-1}+q}ight] \ & =\frac{(q-1)^2}{(q-1)^2+p^2}\left[\frac{p^2+p^2(q-1)+q(q-1)^2}{(q-1)^2}ight] \end{aligned} $

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2+p x+q$ $=0$, then the value of $\sin…

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