midtlinjeg

2021-09-05

$\mathrm{cot}2\left(\mathrm{sin}\left(\theta \right)\right)$
Find the derivative.

Usamah Prosser

$y=\mathrm{cot}2\left(\mathrm{sin}\left(\theta \right)\right)$
$\frac{dy}{dth\eta }=\frac{d}{dth\eta }\left({\mathrm{cot}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)\right)$
$=2\mathrm{cot}\left(\mathrm{sin}\left(\theta \right)\right)×\frac{d}{dth\eta }\left(\mathrm{cot}\left(\mathrm{sin}\left(\theta \right)\right)\right)$
$=2\mathrm{cot}\left(\mathrm{sin}\left(\theta \right)\right)×\left\{-{\mathrm{csc}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)\right\}×\frac{d}{dth\eta }\left(\mathrm{sin}\theta \right)$
$=-2\mathrm{cot}\left(\mathrm{sin}\left(\theta \right)\right)×-{\mathrm{csc}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)×\mathrm{cos}\theta$
$=-2{\mathrm{csc}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)×{\mathrm{csc}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)×\mathrm{cos}\theta$
$=-2{\mathrm{csc}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)×\mathrm{cot}\left(\mathrm{sin}\left(\theta \right)\right)×\mathrm{cos}\theta$
Thus $\frac{dy}{dth\eta }=-2{\mathrm{csc}}^{2}\left(\mathrm{sin}\left(\theta \right)\right)×\mathrm{cot}\left(\mathrm{sin}\left(\theta \right)\right)×\mathrm{cos}\theta$

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