For theta in left(0, frac{pi}{2}right),operat | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We know that the inverse hyperbolic secant function is defined as $\operatorname{sech}^{-1}(x) = \log \left( \frac{1 + \sqrt{1 - x^2}}{x} \right)$

Vedclass · Accepted Answer

$\log \left|	an \left(\frac{\pi}{4}+\frac{	heta}{2}ight)ight|$

Vedclass · Answer

(C) We know that the inverse hyperbolic secant function is defined as $\operatorname{sech}^{-1}(x) = \log \left( \frac{1 + \sqrt{1 - x^2}}{x} ight)$.Substituting $x = \cos 	heta$,we get:$\operatorname{sech}^{-1}(\cos 	heta) = \log \left( \frac{1 + \sqrt{1 - \cos^2 	heta}}{\cos 	heta} ight)$Since $	heta \in \left(0, \frac{\pi}{2}ight)$,$\sin 	heta > 0$,so $\sqrt{1 - \cos^2 	heta} = \sin 	heta$.Thus,the expression becomes $\log \left( \frac{1 + \sin 	heta}{\cos 	heta} ight)$.This can be rewritten as $\log (\sec 	heta + 	an 	heta)$.Using the identity $\sec 	heta + 	an 	heta = 	an \left( \frac{\pi}{4} + \frac{	heta}{2} ight)$,we obtain:$\operatorname{sech}^{-1}(\cos 	heta) = \log \left| 	an \left( \frac{\pi}{4} + \frac{	heta}{2} ight) ight|$.

For $\theta \in \left(0, \frac{\pi}{2}\right)$,$\operatorname{sech}^{-1}(\cos \theta)$ is equal to

Similar Questions

Let $f(x) = \cos^{-1}\left(\frac{2x}{1+x^2}\right) + \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$,then the value of $f(1) + f(2)$ is -

Considering only the principal values of inverse trigonometric functions,the number of positive real values of $x$ satisfying $\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is:

The value of $\tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)$ is

If $\sin ^{-1}\left(\frac{2 a}{1+a^2}\right)+\cos ^{-1}\left(\frac{1-a^2}{1+a^2}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)$ where $a, x \in(0,1)$,then the value of $x$ is

If $\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2},$ then the value of $x^{9}+y^{9}+z^{9}-\frac{1}{x^{9} y^{9} z^{9}}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module