mathop {lim }limits_{theta to pi /2} (sec the | vedclass

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(A) We need to evaluate $\mathop {\lim }\limits_{	heta 	o \pi /2} (\sec 	heta - 	an 	heta )$.Substituting $\sec 	heta = \frac{1}{\cos 	heta}$ a

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(A) We need to evaluate $\mathop {\lim }\limits_{	heta 	o \pi /2} (\sec 	heta - 	an 	heta )$.Substituting $\sec 	heta = \frac{1}{\cos 	heta}$ and $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$,we get:$\mathop {\lim }\limits_{	heta 	o \pi /2} \left( \frac{1 - \sin 	heta}{\cos 	heta} ight)$.As $	heta 	o \pi /2$,this is a $\frac{0}{0}$ indeterminate form.Using the identity $1 - \sin 	heta = \left( \cos \frac{	heta}{2} - \sin \frac{	heta}{2} ight)^2$ and $\cos 	heta = \cos^2 \frac{	heta}{2} - \sin^2 \frac{	heta}{2} = (\cos \frac{	heta}{2} - \sin \frac{	heta}{2})(\cos \frac{	heta}{2} + \sin \frac{	heta}{2})$:$\mathop {\lim }\limits_{	heta 	o \pi /2} \frac{(\cos \frac{	heta}{2} - \sin \frac{	heta}{2})^2}{(\cos \frac{	heta}{2} - \sin \frac{	heta}{2})(\cos \frac{	heta}{2} + \sin \frac{	heta}{2})} = \mathop {\lim }\limits_{	heta 	o \pi /2} \frac{\cos \frac{	heta}{2} - \sin \frac{	heta}{2}}{\cos \frac{	heta}{2} + \sin \frac{	heta}{2}}$.Substituting $	heta = \pi / 2$,we get $\frac{\cos(\pi/4) - \sin(\pi/4)}{\cos(\pi/4) + \sin(\pi/4)} = \frac{1/\sqrt{2} - 1/\sqrt{2}}{1/\sqrt{2} + 1/\sqrt{2}} = \frac{0}{\sqrt{2}} = 0$.

$\mathop {\lim }\limits_{\theta \to \pi /2} (\sec \theta - \tan \theta ) = $

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