If tan x = (1 – cos y)/sin y, Then Find tan 2x
Question:
\[ \tan x=\frac{1-\cos y}{\sin y} \]Find the value of \(\tan 2x\).
Solution
Using the standard half-angle identity
\[ \tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta} \]Comparing with the given expression,
\[ \tan x = \frac{1-\cos y}{\sin y} = \tan\frac{y}{2} \]Therefore,
\[ x=\frac{y}{2} \]Multiplying both sides by 2,
\[ 2x=y \]Hence,
\[ \tan 2x=\tan y \]