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 within the inverse tangent function.
The expression to simplify is:\[\tan^-1 \left( \frac\cos x1 - \sin x \right).\]Apply trigonometric identities to rewrite \( 1 - \sin x \) as \( (\cos\fracx2 - \sin\fracx2)^2 \).
Step 2: Convert the expression to a single tangent function.
Substitute \( 1 - \sin x = (\cos\fracx2 - \sin\fracx2)^2 \) and \( \cos x = \cos^2\fracx2 - \sin^2\fracx2 \). This leads to:\[\tan^-1 \left( \frac\cos x1 - \sin x \right)= \tan^-1 \left[ \tan\left(\frac\pi4 + \fracx2\right) \right].\]
Step 3: Simplify using the identity \( \tan^-1 \tan y = y \).
Given the condition \( -\frac\pi2<x<\frac\pi2 \), the expression simplifies to:\[\tan^-1 \left[ \tan\left(\frac\pi4 + \fracx2\right) \right] = \frac\pi4 + \fracx2.\]
Conclusion: The simplified form of the expression is \( \frac\pi4 + \fracx2 \).