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 :

(A), (C), (D) Only

$(A)\quad \tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right)$ 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).