Question

If $t=\frac{\left(}{1-\mathrm{cos\theta }}\mathrm{and}s=2\tan \left(\frac{\theta }{2}\right)$, then find $\frac{\mathrm{dt}}{\mathrm{ds}}\mathrm{at}\theta =\frac{\pi }{2}$

• A. 2
• B. $\frac{1}{2}$
• C. \N
• D. 1

Answer 1

The correct option is D 1

$t=\frac{\left(}{1-\mathrm{cos\theta }}=\frac{2{\sin }^2\frac{\theta }{2}}{2{\cos }^2\frac{\theta }{2}}={\tan }^2\frac{\theta }{2}\mathrm{and}s=2\tan \left(\frac{\theta }{2}\right)\frac{\mathrm{dt}}{\mathrm{d\theta }}=2\left(\tan \frac{\theta }{2}{\sec }^2\frac{\theta }{2}\right)\frac{1}{2}\frac{\mathrm{ds}}{\mathrm{d\theta }}=2\left({\sec }^2\frac{\theta }{2}\right)\frac{1}{2}{\frac{\mathrm{dt}}{\mathrm{ds}}}_{\left(\theta =\frac{\pi }{2}\right)}=\frac{\frac{\mathrm{dt}}{\mathrm{d\theta }}}{\frac{\mathrm{ds}}{\mathrm{d\theta }}}={\tan \frac{\theta }{2}}_{\left(\theta =\frac{\pi }{2}\right)}=1$