Which of the following statements are correct ?
(A) $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy }\right), xy < 1 $
(B) $\tan^{-1}x+\tan^{-1}y = \pi + \tan^{-1}\left(\frac{x+y}{1-xy }\right), xy > 1 $
(C) $\sin^{-1}x+\cos^{-1}x=\frac{\pi }{2} , x \in (-1, 1)$
(D) $\tan^{-1}x+\cot^{-1}x=\frac{\pi }{2} , x \in R$
(E) $\cos^{-1}(-x) =-\cos^{-1}x, x \in [-1, 1]$
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 1
$(A)\quad \tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right)$ holds only when $xy<1$ and sum lies in principal range
$(A)\ \text{is correct}$
$(B)\quad \tan^{-1}x+\tan^{-1}y=\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\ \text{is not generally valid for }xy>1$
$(B)\ \text{is false}$
$(C)\quad \sin^{-1}x+\cos^{-1}x=\frac{\pi}{2},\ x\in[-1,1]$
Given $x\in(-1,1)$, relation holds
$(C)\ \text{is correct}$
$(D)\quad \tan^{-1}x+\cot^{-1}x=\frac{\pi}{2},\ x\in\mathbb{R}$
$(D)\ \text{is correct}$
$(E)\quad \cos^{-1}(-x)=\pi-\cos^{-1}x\neq -\cos^{-1}x$
$(E)\ \text{is false}$
Correct statements are (A), (C), (D).