Question
Complete the identity. \( \frac{1-\cos x}{\sin x}=? \)
Ask by Hardy Turner. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{1 - \cos x}{\sin x} = \tan\left(\frac{x}{2}\right)
\]
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Extra Insights
To complete the identity, recognize that \( \frac{1 - \cos x}{\sin x} \) can be simplified using the half-angle identity. Specifically, this expression simplifies to \( \tan\left(\frac{x}{2}\right) \). Why? Because using the half-angle formulas, we have: \[ \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} \] and with a bit of manipulation involving the Pythagorean identity, you can get to \( \frac{1 - \cos x}{\sin x} \). So, the final answer is \( \tan\left(\frac{x}{2}\right) \). Enjoy your trigonometric explorations!
