Find `(dy)/(dx)` , when `x=cos^(-1)1/(sqrt(1+t^2))` and `y=sin^(-1)t/(sqrt(1+t^2)),\ \ t in R`

Explanation

To find \(\frac{dy}{dx}\) when \(x = \cos^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right)\) and \(y = \sin^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right)\), we can follow these steps:

### Step 1: Substitute \(t\) with \(\tan(\theta)\)
Let’s set \(t = \tan(\theta)\). This substitution is useful because it simplifies the expressions involving \(t\).

### Step 2: Simplify \(x\)
Using the substitution, we rewrite \(x\):
\[
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