If x=log _eleft[cot left(frac{pi}{4}+thetarig | vedclass

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

Question

(C) We have $x=\log _e \left[\cot \left(\frac{\pi}{4}+	hetaight)ight]$.For statement $I$:$\cosh x = \frac{e^x+e^{-x}}{2} = \frac{\cot \left(\frac

Vedclass · Accepted Answer

Both $I$ and $II$ are true

Vedclass · Answer

(C) We have $x=\log _e \left[\cot \left(\frac{\pi}{4}+	hetaight)ight]$.For statement $I$:$\cosh x = \frac{e^x+e^{-x}}{2} = \frac{\cot \left(\frac{\pi}{4}+	hetaight) + 	an \left(\frac{\pi}{4}+	hetaight)}{2}$$= \frac{\frac{\cos(\pi/4+	heta)}{\sin(\pi/4+	heta)} + \frac{\sin(\pi/4+	heta)}{\cos(\pi/4+	heta)}}{2} = \frac{\cos^2(\pi/4+	heta) + \sin^2(\pi/4+	heta)}{2 \sin(\pi/4+	heta) \cos(\pi/4+	heta)}$$= \frac{1}{\sin(2(\pi/4+	heta))} = \frac{1}{\sin(\pi/2+2	heta)} = \frac{1}{\cos 2	heta} = \sec 2	heta$.Thus,statement $I$ is true.For statement $II$:$\sinh x = \frac{e^x-e^{-x}}{2} = \frac{\cot \left(\frac{\pi}{4}+	hetaight) - 	an \left(\frac{\pi}{4}+	hetaight)}{2}$$= \frac{\frac{\cos^2(\pi/4+	heta) - \sin^2(\pi/4+	heta)}{\sin(\pi/4+	heta) \cos(\pi/4+	heta)}}{2} = \frac{\cos(2(\pi/4+	heta))}{\sin(2(\pi/4+	heta))}$$= \cot(\pi/2+2	heta) = -	an 2	heta$.Thus,statement $II$ is true.

If $x=\log _e\left[\cot \left(\frac{\pi}{4}+\theta\right)\right]$ and $\theta \in\left(\frac{-\pi}{4}, \frac{\pi}{4}\right)$,then consider the following statements:
$I$. $\cosh x=\sec 2 \theta$
$II$. $\sinh x=-\tan 2 \theta$

Similar Questions

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is

If $(\sec \alpha + \tan \alpha )(\sec \beta + \tan \beta )(\sec \gamma + \tan \gamma ) = \tan \alpha \tan \beta \tan \gamma $,then $(\sec \alpha - \tan \alpha )(\sec \beta - \tan \beta )(\sec \gamma - \tan \gamma ) = $

The number of elements in the set $S = \{\theta \in [-4\pi, 4\pi] : 3 \cos^2 2\theta + 6 \cos 2\theta - 10 \cos^2 \theta + 5 = 0\}$ is

If $x$ and $y$ are acute angles such that $\cos x + \cos y = \frac{3}{2}$ and $\sin x + \sin y = \frac{3}{4}$,then $\sin(x + y)$ equals to

The smallest positive value of $x$ (in degrees) for which $\tan(x+100^{\circ}) = \tan(x+50^{\circ}) \tan(x) \tan(x-50^{\circ})$ is (in $^{\circ}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module

If $x=\log _e\left[\cot \left(\frac{\pi}{4}+\theta\right)\right]$ and $\theta \in\left(\frac{-\pi}{4}, \frac{\pi}{4}\right)$,then consider the following statements:$I$. $\cosh x=\sec 2 \theta$$II$. $\sinh x=-\tan 2 \theta$