Question

A value of \(x\) satisfying the equation \(\sin\left[\cot^{-1}(1+x)\right]=\cos\left[\tan^{-1}x\right]\), is :

• A. \(-\dfrac12\)
• B. \(-1\)
• C. \(0\)
• D. \(\dfrac12\)

Hint:

\(\cot^{-1}(1+x)\in(0,\pi)\), so \(\sin\left[\cot^{-1}(1+x)\right]=\frac1{\sqrt{1+(1+x)^2}}\).
\(\cos[\tan^{-1}x]=\frac1{\sqrt{1+x^2}}\).
Equate: \(1+(1+x)^2=1+x^2\Rightarrow 1+2x=0\Rightarrow x=-\frac12\).
Answer: (1) \(-\frac12\)

Answer 1

The Correct Option is A

Express both sides algebraically: \(\sin(\cot^{-1}u)=\frac1{\sqrt{1+u^2}}\) and \(\cos(\tan^{-1}x)=\frac1{\sqrt{1+x^2}}\).