Question

Evaluate $\underset{\theta \to \frac{\pi }{2}}{\lim }$ $\left(sec\theta -tan\theta \right)$ is

• A. $0$
• B. $1$
• C. $4$
• D. Not defined

Answer 1

Answer: (A)

$\underset{\theta \to \frac{\pi }{2}}{\lim }$ $\left(sec\theta -tan\theta \right)$
$=\underset{\theta \to \frac{\pi }{2}}{\lim }$$\left(\frac{1}{cos\theta }-\frac{sin\theta }{cos\theta }\right)$
$=\underset{\theta \to \frac{\pi }{2}}{\lim }$$\frac{1-sin\theta }{cos\theta }$
$=\underset{\theta \to \frac{\pi }{2}}{\lim }$$\frac{co{s}^2\frac{\theta }{2}+si{n}^2\frac{\theta }{2}-2\cdot sin\frac{\theta }{2}\cdot cos\frac{\theta }{2}}{co{s}^2\frac{\theta }{2}-si{n}^2\frac{\theta }{2}}$
$=\underset{\theta \to \frac{\pi }{2}}{\lim }$$\left[\frac{{\left(cos\frac{\theta }{2}-sin\frac{\theta }{2}\right)}^2}{\left(cos\frac{\theta }{2}+sin\frac{\theta }{2}\right)\left(cos\frac{\theta }{2}-sin\frac{\theta }{2}\right)}\right]$
$=\underset{\theta \to \frac{\pi }{2}}{\lim }$$\frac{cos\frac{\theta }{2}-sin\frac{\theta }{2}}{cos\frac{\theta }{2}+sin\frac{\theta }{2}}=\frac{cos\frac{\pi }{4}-sin\frac{\pi }{4}}{cos\frac{\pi }{4}+sin\frac{\pi }{4}}$
$=\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=0$