Question

Evaluate:
$(sec\theta -cos\theta )(cos\theta +tan\theta )$

Answer 1

$(\sec \theta -\cos \theta )(\cos \theta +\tan \theta )$

$=\sec \theta (\cos \theta +\tan \theta )-\cos \theta (\cos \theta +\tan \theta )$

$=\sec \theta \cos \theta +\sec \theta \tan \theta -{\cos }^2\theta -\cos \theta \tan \theta$

$=1+\sin \theta {\sec }^2\theta -{\cos }^2\theta -\sin \theta$

$=1-{\cos }^2\theta +\sin \theta {\sec }^2\theta -\sin \theta$

$={\sin }^2\theta +\sin \theta {\sec }^2\theta -\sin \theta$

$={\sin }^2\theta +\sin \theta ({\sec }^2\theta -1)$

$={\sin }^2\theta +\sin \theta \left({\tan }^2\theta \right)$

$={\sin }^2\theta +\sin \theta \frac{{\sin }^2\theta }{{\cos }^2\theta }$

$={\sin }^2\theta (1+\tan \theta \sec \theta )$