Question

$\int \frac{\cos ec\theta -\cot \theta }{\cos ec\theta +\cot \theta }d\theta =$

• A. $2\cos ec\theta -2\cot \theta -\theta +c$
• B. $2\cos ec\theta -2\cot \theta +\theta +c$
• C. $2\cos ec\theta +2\cot \theta -\theta +c$
• D. NONE OF THE ABOVE

Answer 1

The correct option is A $2\cos ec\theta -2\cot \theta -\theta +c$

$\begin{array}{c}2\tan \theta h-\theta +c\\ =2\left(\frac{\sin \theta h}{\cos \theta h}\right)-\theta +c\\ =\frac{2\sin \theta h\times \cos \theta h}{{\cos }^2\theta h}-\theta +c\\ =\frac{\sin \theta }{\left(\frac{1+\cos \theta }{2}\right)}-\theta +c\\ =\frac{2\sin \theta \left(1-\cos \theta \right)}{\left(1+\cos \theta \right)\left(1-\cos \theta \right)}-\theta +c\\ =\frac{2\sin \theta \left(1-\cos \theta \right)}{{\sin }^2\theta }-\theta +c\\ =2\left(\frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta }\right)-\theta +c\\ =2\left(\cos ec\theta -\cot \theta \right)-\theta +c\end{array}$