Differentiate the following :
(iii) `-(tan theta)/(tan theta+sec theta)`

Explanation

To differentiate the expression \(-\frac{\tan \theta}{\tan \theta + \sec \theta}\), we will apply the quotient rule of differentiation. The quotient rule states that if you have a function in the form of \(\frac{u}{v}\), then its derivative is given by:

\[ \frac{d}{d\theta}\left(\frac{u}{v}\right) = \frac{v \frac{du}{d\theta} - u \frac{dv}{d\theta}}{v^2} \]

Here, we will identify:
- \(u = -\tan \theta\)
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