If tan A = (1 − cos B)/sin B, Then Find tan 2A

Question

If

\[ \tan A=\frac{1-\cos B}{\sin B}, \]

then find the value of

\[ \tan 2A. \]

Solution

Using the standard half-angle identity

\[ \tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}, \]

we get

\[ \tan A = \tan\frac{B}{2}. \]

Therefore,

\[ A=\frac{B}{2}. \]

Multiplying both sides by 2,

\[ 2A=B. \]

Hence,

\[ \tan 2A = \tan B. \]

Answer

\[ \boxed{\tan B} \]