Question

If $\sec \theta +\tan \theta =h$, then $\frac{h^2-1}{h^2+1}=\sin \theta$.

• A. True
• B. False

Answer 1

The correct option is A True

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

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

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

$=\frac{{\tan }^2\theta +{\tan }^2\theta +2\sec \theta \tan \theta }{{\sec }^2\theta +{\sec }^2\theta +2\sec \theta \tan \theta }$

$=\frac{2{\tan }^2\theta +2\sec \theta \tan \theta }{2{\sec }^2\theta +2\sec \theta \tan \theta }$

$=\frac{{\tan }^2\theta +\sec \theta \tan \theta }{{\sec }^2\theta +\sec \theta \tan \theta }$

$=\frac{\tan \theta (\tan \theta +\sec \theta )}{\sec \theta (\tan \theta +\sec \theta )}$

$=\frac{\tan \theta }{\sec \theta }$

$=\frac{\sin \theta }{\cos \theta }\times \cos \theta =\sin \theta$