Question

Express \( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \), where \( -\frac{\pi}{2}<x<\frac{\pi}{2} \), 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:\[\tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right).\]To simplify, rewrite \( 1 - \sin x \) using the identity \( 1 = \cos^2\frac{x}{2} + \sin^2\frac{x}{2} \) and \( \sin x = 2\sin\frac{x}{2}\cos\frac{x}{2} \):\[1 - \sin x = (\cos^2\frac{x}{2} + \sin^2\frac{x}{2}) - 2\sin\frac{x}{2}\cos\frac{x}{2} = (\cos\frac{x}{2} - \sin\frac{x}{2})^2.\]Step 2: {Transform into a single tangent function}
Substitute \( 1 - \sin x = (\cos\frac{x}{2} - \sin\frac{x}{2})^2 \) and \( \cos x = \cos^2\frac{x}{2} - \sin^2\frac{x}{2} \). Then, simplify the fraction:\[\frac{\cos x}{1 - \sin x} = \frac{\cos^2\frac{x}{2} - \sin^2\frac{x}{2}}{(\cos\frac{x}{2} - \sin\frac{x}{2})^2} = \frac{(\cos\frac{x}{2} - \sin\frac{x}{2})(\cos\frac{x}{2} + \sin\frac{x}{2})}{(\cos\frac{x}{2} - \sin\frac{x}{2})^2} = \frac{\cos\frac{x}{2} + \sin\frac{x}{2}}{\cos\frac{x}{2} - \sin\frac{x}{2}}.\]Divide the numerator and denominator by \( \cos\frac{x}{2} \):\[\frac{1 + \tan\frac{x}{2}}{1 - \tan\frac{x}{2}}.\]Using the tangent addition formula \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \), where \( \tan\frac{\pi}{4} = 1 \):\[\frac{\tan\frac{\pi}{4} + \tan\frac{x}{2}}{1 - \tan\frac{\pi}{4} \tan\frac{x}{2}} = \tan\left(\frac{\pi}{4} + \frac{x}{2}\right).\]Thus, the expression becomes:\[\tan^{-1} \left[ \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) \right].\]Step 3: {Simplify using \( \tan^{-1} \tan y = y \)}
Given that \( -\frac{\pi}{2}<x<\frac{\pi}{2} \), it follows that \( -\frac{\pi}{4}<\frac{x}{2}<\frac{\pi}{4} \), and therefore \( 0<\frac{\pi}{4} + \frac{x}{2}<\frac{\pi}{2} \). This range is within the principal value range of \( \tan^{-1} \tan y = y \). Hence, we simplify:\[\tan^{-1} \left[ \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) \right] = \frac{\pi}{4} + \frac{x}{2}.\]Conclusion: The simplest form is \( \frac{\pi}{4} + \frac{x}{2} \).