Question

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

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

Answer 1

The correct option is C $\frac{\pi }{2}$

$\because \theta ={tan}^{-1}a\Rightarrow a=tan\theta and\varphi ={tan}^{-1}b\Rightarrow b=tan\varphi alsoab=-1\Rightarrow tan\theta .tan\varphi =-1\Rightarrow \frac{sin\theta }{cos\theta }.\frac{sin\varphi }{cos\varphi }=-1\Rightarrow sin\theta .sin\varphi =-cos\theta .cos\varphi \Rightarrow cos\theta .cos\varphi +sin\theta .sin\varphi =0\Rightarrow cos(\theta -\varphi )=cos\frac{\pi }{2}\Rightarrow \theta -\varphi =\frac{\pi }{2}$