Question

If $\theta ={\tan }^{-1}a$, $\varphi ={\tan }^{-1}b$ and $ab=-1$, then $\theta -\varphi =$

• A. $0$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{2}$
• D. None of these

Answer 1

The correct option is C

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct answer:

Finding the value of $\theta -\varphi$:

$$\begin{gathered} \theta ={\tan }^{-1}a \\ \Rightarrow a=\tan \theta \end{gathered}$$

$$\begin{gathered} \varphi ={\tan }^{-1}b \\ \Rightarrow b=\tan \varphi \end{gathered}$$

Substitute value of $a$and $b$in $ab=-1$

$$\begin{gathered} ab=-1 \\ \Rightarrow \tan \theta .\tan \varphi =-1 \\ \Rightarrow \tan \theta =\frac{-1}{\tan \varphi } \\ \Rightarrow \tan \theta =-cot\varphi \\ \Rightarrow \tan \theta =cot(-\varphi ) \\ \Rightarrow \tan \theta =\tan \left(\frac{\mathrm{\pi }}{2}+\varphi \right)\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{c}\mathbf{o}\mathbf{t}\mathbf{(}\mathbf{-}\mathbf{A}\mathbf{)}\mathbf{=}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{(}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{+}\mathbf{A}\mathbf{)}\mathbf{]} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{2}+\varphi \\ \Rightarrow \theta -\varphi =\frac{\mathrm{\pi }}{2} \end{gathered}$$

Hence, Option (C) is the correct answer.