If y()=2(+)1+^2+1^2, (34,), then dyd at =56 is:

Question

If $y(\alpha)=\dfrac{2(	an\alpha+\cot\alpha)}{1+	an^2\alpha}+\dfrac{1}{\sin^2\alpha}$, $\alpha\in\left(\dfrac{3\pi}{4},\piight)$, then $\dfrac{dy}{d\alpha}$ at $\alpha=\dfrac{5\pi}{6}$ is:

Accepted Answer

$y(\alpha)=2\cot\alpha+\csc^2\alpha=2\cot\alpha+1+\cot^2\alpha=(1+\cot\alpha)^2$, so $y=|1+\cot\alpha|=-1-\cot\alpha$ for $\alpha\in\left(\frac{3\pi}4,\pi\right)$ (where $\cot\alpha<-1$).

$\dfrac{dy}{d\alpha}=\csc^2\alpha$; at $\alpha=\dfrac{5\pi}6$, $\csc\dfrac{5\pi}6=2\Rightarrow\csc^2=4$.

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