Verified Answer :

  • Let \(\theta = \csc^{-1} u\), implying \(\sin \theta = 1/u\).
  • Use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\) to get \(\cos^2 \theta = 1 - 1/u^2 = (u^2-1)/u^2\).
  • Take the positive square root because \(\cos \theta \ge 0\) for \(\theta \in [-\pi/2, \pi/2]\).
  • The simplified expression is \(\boxed{\frac{\sqrt{u^2 - 1}}{|u|}}\).

Explanation

  1. Defining the variable and identifying the domain

To simplify the expression \(\cos(\csc^{-1} u)\), we define \(\theta = \csc^{-1} u\). By the definition of the inverse cosecant function, this implies that \(\csc \theta = u\), where the range of \(\theta\) is \([-\pi/2, 0) \cup (0, \pi/2]\) and the domain of \(u\) is \(|u| \geq 1\).

  1. Using trigonometric identities

We know that \(\csc \theta = \frac{1}{\sin \theta} = u\), which leads to \(\sin \theta = \frac{1}{u}\). Using the fundamental trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\), we can express \(\cos^2 \theta\) as:
\[\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{1}{u}\right)^2 = 1 - \frac{1}{u^2} = \frac{u^2 - 1}{u^2}.\]

  1. Determining the sign and final simplification

Taking the square root, we have \(\cos \theta = \pm \sqrt{\frac{u^2 - 1}{u^2}} = \pm \frac{\sqrt{u^2 - 1}}{|u|}\). Since \(\theta\) is restricted to the interval \([-\pi/2, 0) \cup (0, \pi/2]\), the cosine of \(\theta\) must be non-negative. Therefore, we select the positive root. The simplified expression is \(\frac{\sqrt{u^2 - 1}}{|u|}\).

Examples

This simplification is useful in calculus and physics when dealing with inverse trigonometric substitutions. For example, if you are integrating a function involving \(\sqrt{x^2 - 1}\) over \(x^2\), you might substitute \(x = \csc \theta\). Simplifying expressions like \(\cos(\csc^{-1} u)\) helps convert complex trigonometric integrals back into algebraic ones, making them much easier to evaluate.

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