# How do you differentiate f(x)= (1 - sin^2x)/(cosx-1)  using the quotient rule?

Jul 31, 2017

see below

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(\frac{1 - {\sin}^{2} \left(x\right)}{\cos \left(x\right) - 1}\right) = \frac{d}{\mathrm{dx}} \left(\frac{{\cos}^{2} \left(x\right)}{\cos \left(x\right) - 1}\right)$

$\implies \frac{\left(\cos \left(x\right) - 1\right) \cdot \frac{d}{\mathrm{dx}} \left({\cos}^{2} \left(x\right)\right) - {\cos}^{2} \left(x\right) \cdot \frac{d}{\mathrm{dx}} \left(\cos \left(x\right) - 1\right)}{\cos \left(x\right) - 1} ^ 2$

$\implies \frac{\sin \left(x\right) \cdot {\cos}^{2} \left(x\right) - 2 \cdot \sin \left(x\right) \cos \left(x\right) \cdot \left(\cos \left(x\right) - 1\right)}{\cos \left(x\right) - 1} ^ 2$

$\implies \frac{\sin \left(x\right) \cos \left(x\right) \cdot \left(2 - \cos \left(x\right)\right)}{\cos \left(x\right) - 1} ^ 2$
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