Question

Simplify and solve:

$\frac{\sin \theta }{(\sec \theta +\tan \theta -1)}+\frac{\cos \theta }{cosec\theta +\cot \theta -1}$.

Answer 1

$\frac{\sin \theta }{(\sec \theta +\tan \theta -1)}+\frac{\cos \theta }{(\csc \theta +\cot \theta -1)}$
$=\frac{\sin \theta (\csc \theta +\cot \theta -1)+\cos \theta (\sec \theta +\tan \theta -1)}{(\sec \theta +\tan \theta -1)(\csc \theta +\cot \theta -1)}$
$=\frac{(1+\cos \theta -\sin \theta +1+\sin \theta -\cos \theta )}{(\sec \theta \csc \theta +\csc \theta -\sec \theta +\sec \theta +1-\tan \theta -\csc \theta -\cot \theta +1)}$
$=\frac{2}{\sec \theta \csc \theta -\tan \theta -\cot \theta +2}$
$=\frac{2}{\sec \theta \csc \theta -(\tan \theta +\frac{1}{\tan \theta })+2}$

$=\frac{2}{\sec \theta \csc \theta -\left(\frac{{\tan }^2\theta +1}{\tan \theta }\right)+2}$
$=\frac{2}{\sec \theta \csc \theta -\left(\frac{{\sec }^2\theta }{\tan \theta }\right)+2}$

$=\frac{2}{\sec \theta \csc \theta -\sec \theta \csc \theta +2}$
$=\frac{2}{2}=1$