Difference between revisions of "Derivative of cosecant"
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where $\csc$ denotes the [[cosecant]] function and $\cot$ denotes the [[cotangent]] function. | where $\csc$ denotes the [[cosecant]] function and $\cot$ denotes the [[cotangent]] function. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> █ | + | <strong>Proof:</strong> Using the [[product rule]] and the definitions of [[cosecant]] and [[cotangent]], |
| + | $$\begin{array}{ll} | ||
| + | \dfrac{\mathrm{d}}{\mathrm{d}z} \csc(z) &= \dfrac{\mathrm{d}}{\mathrm{d}z} \left[ \dfrac{1}{\sin(z)} \right] \\ | ||
| + | &= \dfrac{0-\cos(z)}{\sin^2(z)} \\ | ||
| + | &= \csc(z)\cot(z), | ||
| + | \end{array}$$ | ||
| + | as was to be shown. █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 04:28, 8 February 2016
Proposition: The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \csc(z)=- \cot(z)\csc(z),$$ where $\csc$ denotes the cosecant function and $\cot$ denotes the cotangent function.
Proof: Using the product rule and the definitions of cosecant and cotangent, $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \csc(z) &= \dfrac{\mathrm{d}}{\mathrm{d}z} \left[ \dfrac{1}{\sin(z)} \right] \\ &= \dfrac{0-\cos(z)}{\sin^2(z)} \\ &= \csc(z)\cot(z), \end{array}$$ as was to be shown. █
