Inverse Trigonometric Functions Question 180
Question: If $ \sin ({{\cot }^{-1}}(x+1)=\cos ({{\tan }^{-1}}x) $ , then x =
[IIT Screening 2004]
Options:
A) $ -\frac{1}{2} $
B) $ \frac{1}{2} $
C) 0
D) $ \frac{9}{4} $
Show Answer
Answer:
Correct Answer: A
Solution:
$ \sin [{{\cot }^{-1}}(x+1)]=\sin ( {{\sin }^{-1}}\frac{1}{\sqrt{x^{2}+2x+2}} ) $
$ =\frac{1}{\sqrt{x^{2}+2x+2}} $
$ \cos ({{\tan }^{-1}}x)=\cos ( {{\cos }^{-1}}\frac{1}{\sqrt{1+x^{2}}} )=\frac{1}{\sqrt{1+x^{2}}} $ Thus, $ \frac{1}{\sqrt{x^{2}+2x+2}}=\frac{1}{\sqrt{1+x^{2}}} $
$ \Rightarrow x^{2}+2x+2=1+x^{2} $
$ \Rightarrow $ $ x=-\frac{1}{2} $ .
BETA
