Question

Express \( \tan^-1 \left( \frac\cos x1 - \sin x \right) \), where \( -\frac\pi2<x<\frac\pi2 \), in the simplest form.

Hint:

For expressions involving \( \tan^-1 \), rewrite in terms of trigonometric identities to simplify.

Answer 1

Step 1: Simplify the expression inside \( \tan^-1 \)
The given expression is: \[ \tan^-1 \left( \frac\cos x1 - \sin x \right). \] Using trigonometric identities, rewrite: \[ 1 - \sin x = (\cos^2\fracx2 + \sin^2\fracx2) - 2\sin\fracx2\cos\fracx2 = (\cos\fracx2 - \sin\fracx2)^2. \]
Step 2: Transform into a single tangent function
Substituting \( 1 - \sin x \) and \( \cos x = (\cos^2\fracx2 - \sin^2\fracx2) \): \[ \tan^-1 \left( \frac\cos x1 - \sin x \right) = \tan^-1 \left[ \tan\left(\frac\pi4 + \fracx2\right) \right]. \]
Step 3: Simplify using \( \tan^-1 \tan y = y \)
Since \( -\frac\pi2<x<\frac\pi2 \), we simplify: \[ \tan^-1 \left[ \tan\left(\frac\pi4 + \fracx2\right) \right] = \frac\pi4 + \fracx2. \]
Conclusion: The simplest form is \( \frac\pi4 + \fracx2 \).