Derivative of \( \displaystyle \ln{\left(\tan{\left(\frac{x}{2} \right)} \right)} \)

Problem 2.2167 · hard

1. \[ \frac{d}{d x} \ln{\left(\tan{\left(\frac{x}{2} \right)} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)}} \]
   chainApply the chain rule for the logarithm.✓ Proved
3. \[ = \frac{\sec^{2}{\left(\frac{x}{2} \right)} \frac{d}{d x} \frac{x}{2}}{\tan{\left(\frac{x}{2} \right)}} \]
   chainApply the chain rule for the tangent function.✓ Proved
4. \[ = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} \]
   derivativeDifferentiate the inner function x/2.✓ Proved
5. \[ = \frac{1}{2 \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}} \]
   algebraMultiply the terms together.✓ Proved
6. \[ = \frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} \]
   rewrite algebraRewrite tangent in terms of sine and cosine. Simplify the denominator.✓ Proved
7. \[ = \frac{1}{\sin{\left(x \right)}} \]
   simplifyUse the double angle identity 2*sin(x/2)*cos(x/2) = sin(x).✓ Proved
8. \[ = \csc{\left(x \right)} \]
   simplifyRewrite 1/sin(x) as csc(x).✓ Proved