A value of x satisfying the equation [^-1(1+x)]=[^-1x], is :

Question

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

Accepted Answer

Let $\cot\lambda=1+x$ and $\tan\beta=x$; the equation $\sin\lambda=\cos\beta$ gives

$$\frac{1}{\sqrt{(1+x)^2+1}}=\frac{1}{\sqrt{1+x^2}}\ \Rightarrow\ (1+x)^2+1=1+x^2\ \Rightarrow\ 2x+1=0\ \Rightarrow\ x=-\frac12.$$

Solutions · 1