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

If $\sec(x) = \cosh(\theta)$,then find $\tanh^2\left(\frac{\theta}{2}\right)$.

If $\frac{\tan(A-B)}{\tan A} + \frac{\sin^{2}C}{\sin^{2}A} = 1,$ where $A, B, C \in (0, \frac{\pi}{2})$, then:

If $\tan A - \tan B = x$ and $\cot B - \cot A = y,$ then $\cot (A - B) = $

If $\cos \theta - 4 \sin \theta = 1$,then $\sin \theta + 4 \cos \theta$ is equal to

If $15 \sin^{4} \alpha + 10 \cos^{4} \alpha = 6$ for some $\alpha \in R$,then the value of $27 \sec^{6} \alpha + 8 \operatorname{cosec}^{6} \alpha$ is equal to ....... .

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$. $\sinh x=-\tan 2 \theta$