Question #b73a1

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Answer 1 · 2017-09-30T07:28:00

$The Reqd. Exp.= \frac{x}{\sqrt{1 - x^{2}}}, | x | < 1 .$ $"The Reqd. Exp.="x/sqrt(1-x^2), |x| < 1.$

Recall that, $a r c cos x$ $arc cos x$ is defined if, &, only if, $| x | \leq 1 .$ $|x| le 1.$

So, let $a r c cos x = θ, and, | x | \leq 1 .$ $arc cosx=theta, and, |x| le 1.$

$:. costheta=x, and, theta in [0,pi].$

Under the substitution,

The Reqd. Exp. is, $cot theta=costheta/sintheta......(1).$

$costheta=x rArr sin^2theta=1-cos^2theta=1-x^2.$

$:. sintheta=+-sqrt(1-x^2).$

But, $theta in [0,pi], sintheta > 0. :. sintheta==sqrt(1-x^2).$

By $(1)," then, "cottheta=x/sqrt(1-x^2), x!=+-1.$

$rArr cot(arc cos x)=x/sqrt(1-x^2), |x| < 1.$