Question

$\frac{d}{dx}\left({\tan }^{-1}\left(\sec \left(x\right)+\tan \left(x\right)\right)\right)=$

• A. $1$
• B. $\frac{1}{2}$
• C. $\cos \left(x\right)$
• D. $\sec \left(x\right)$

Answer 1

The correct option is B

$\frac{1}{2}$

Explanation for correct option:

Step-1: Simplify the given data.

Given, $\frac{d}{dx}\left({\tan }^{-1}\left(\sec \left(x\right)+\tan \left(x\right)\right)\right)$

$$\begin{gathered} =\frac{d}{dx}\left({\tan }^{-1}\left(\frac{1}{\cos \left(x\right)}+\frac{\sin \left(x\right)}{\cos \left(x\right)}\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\frac{1+\sin \left(x\right)}{\cos \left(x\right)}\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\frac{{\sin }^2\left({\displaystyle \frac{x}{2}}\right)+{\cos }^2\left({\displaystyle \frac{x}{2}}\right)+2\sin \left({\displaystyle \frac{x}{2}}\right)\cos \left({\displaystyle \frac{x}{2}}\right)}{{\cos }^2\left({\displaystyle \frac{x}{2}}\right)-{\sin }^2\left({\displaystyle \frac{x}{2}}\right)}\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\frac{{\left(\sin \left({\displaystyle \frac{x}{2}}\right)+\cos \left({\displaystyle \frac{x}{2}}\right)\right)}^2}{\left(\cos \left({\displaystyle \frac{x}{2}}\right)-\sin \left({\displaystyle \frac{x}{2}}\right)\right)\left(\cos \left({\displaystyle \frac{x}{2}}\right)+\sin \left({\displaystyle \frac{x}{2}}\right)\right)}\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\frac{\left(\sin \left({\displaystyle \frac{x}{2}}\right)+\cos \left({\displaystyle \frac{x}{2}}\right)\right)}{\left(\cos \left({\displaystyle \frac{x}{2}}\right)-\sin \left({\displaystyle \frac{x}{2}}\right)\right)}\right)\right) \end{gathered}$$

Step-2: Divided by $\cos \left(\frac{x}{2}\right)$ both the side.

$$\begin{gathered} =\frac{d}{dx}\left({\tan }^{-1}\left(\frac{1+\tan \left({\displaystyle \frac{x}{2}}\right)}{1-\tan \left({\displaystyle \frac{x}{2}}\right)}\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\tan \left(\frac{x}{2}\right)\right)\right) \\ =\frac{d}{dx}\left(\frac{x}{2}\right) \end{gathered}$$

Step-3: Differentiate w.r.t $x$

$$\begin{gathered} =\frac{d}{dx}\left(\frac{x}{2}\right) \\ =\frac{1}{2}\times \left(1\right) \\ =\frac{1}{2} \end{gathered}$$

Hence, correct answer is option $\left(B\right)$