operatorname{sech}^{-1}(sin theta) is equal t | vedclass

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

Question

(D) Let $y = \operatorname{sech}^{-1}(\sin 	heta)$.Then,$\operatorname{sech} y = \sin 	heta$.Since $\operatorname{sech} y = \frac{1}{\cosh y}$,we ha

Vedclass · Accepted Answer

$\log \cot \frac{	heta}{2}$

Vedclass · Answer

(D) Let $y = \operatorname{sech}^{-1}(\sin 	heta)$.Then,$\operatorname{sech} y = \sin 	heta$.Since $\operatorname{sech} y = \frac{1}{\cosh y}$,we have $\cosh y = \frac{1}{\sin 	heta} = \operatorname{cosec} 	heta$.Thus,$y = \cosh^{-1}(\operatorname{cosec} 	heta)$.Using the logarithmic form of the inverse hyperbolic cosine function,$\cosh^{-1}(x) = \log(x + \sqrt{x^2 - 1})$,we get:$y = \log(\operatorname{cosec} 	heta + \sqrt{\operatorname{cosec}^2 	heta - 1})$.Since $\operatorname{cosec}^2 	heta - 1 = \cot^2 	heta$,we have:$y = \log(\operatorname{cosec} 	heta + \cot 	heta)$.Using the trigonometric identities $\operatorname{cosec} 	heta = \frac{1}{\sin 	heta} = \frac{1}{2 \sin(	heta/2) \cos(	heta/2)}$ and $\cot 	heta = \frac{\cos 	heta}{\sin 	heta} = \frac{\cos^2(	heta/2) - \sin^2(	heta/2)}{2 \sin(	heta/2) \cos(	heta/2)}$,we get:$y = \log\left(\frac{1 + \cos^2(	heta/2) - \sin^2(	heta/2)}{2 \sin(	heta/2) \cos(	heta/2)}ight) = \log\left(\frac{2 \cos^2(	heta/2)}{2 \sin(	heta/2) \cos(	heta/2)}ight) = \log(\cot(	heta/2))$.Therefore,$\operatorname{sech}^{-1}(\sin 	heta) = \log \cot \frac{	heta}{2}$.

$\operatorname{sech}^{-1}(\sin \theta)$ is equal to

Similar Questions

The value of $2(\sin^6 \theta + \cos^6 \theta) - 3(\sin^4 \theta + \cos^4 \theta) + 1$ is

If $f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x}$ for all $x \in R$,then $f(2023) = $

The number of elements in the set $S = \{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right) = 4^{x} + 4^{-x}\}$ is $.....$

Consider the following parametric equation of a curve: $x(\theta) = |\cos 4\theta| \cos \theta$ and $y(\theta) = |\cos 4\theta| \sin \theta$,where $0 \leq \theta \leq 2\pi$. Which one of the following graphs represents the curve?

The graph of the function $f(x) = \cos x \cos(x + 2) - \cos^2(x + 1)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module