Question #b73a1

1 Answer
Sep 30, 2017

Answer:

# "The Reqd. Exp.="x/sqrt(1-x^2), |x| < 1.#

Explanation:

Recall that, #arc cos x# is defined if, &, only if, #|x| le 1.#

So, let #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.#