https://file.aiquickdraw.com/mathbot/file/1727846329664y0ac6l19.jpg

Let's break down the steps shown in the image and simplify the expression.

The original question (labeled as F3) states:
\[
\text{Prove that } \sinh^{-1}(\sin\theta) = \log\left(\cot\frac{\theta}{2}\right).
\]

### Work done in the image:

1. It starts with:
   \[
   \sinh^{-1}(\sin\theta) = \log\left[\sin\theta + \sqrt{\sin^2\theta + 1}\right]
   \]

2. Then simplifies the square root expression:
   \[
   \sqrt{\sin^2\theta + 1} = \cos\theta
   \]
   Since \(\sin^2\theta + \cos^2\theta = 1\), this identity is used to reduce the square root.

3. This gives:
   \[
   \log(\sin\theta + \cos\theta)
   \]

4. The next steps rewrite \(\sin\theta + \cos\theta\) in terms of \( \csc\theta \) and \( \sec\theta \):
   \[
   \log\left(\csc\theta + \sec\theta\right)
   \]

5. Finally, it simplifies into the form:
   \[
   \log\left(\frac{\sec\theta + \csc\theta}{\sec\theta \cdot \csc\theta}\right)
   \]

This final expression still needs further verification or transformation to match \(\log\left(\cot\frac{\theta}{2}\right)\), though the general structure is almost there. One would need to relate this to the half-angle identity for cotangent.

Would you like to dive deeper into this derivation or have more detailed steps? Let me know!

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Here are 5 related questions to expand your understanding:
1. What is the identity for \(\sinh^{-1}(x)\) in terms of logarithms?
2. How does the identity \(\cos^2\theta + \sin^2\theta = 1\) assist in simplifying expressions?
3. What is the half-angle identity for \(\cot(\theta/2)\)?
4. How can \(\csc\theta\) and \(\sec\theta\) be expressed in terms of sine and cosine?
5. What are the key steps to transform \(\sinh^{-1}(\sin\theta)\) into a logarithmic expression?

**Tip:** Always look for trigonometric identities or hyperbolic identities when simplifying complex trigonometric and inverse expressions!

Show that tanh^{-1}(x) = sinh^{-1}(x / sqrt(1 - x^2)) and prove that sinh^{-1}(sinθ) = log(cot(θ/2)).

This page provides a detailed solution to the problem of proving that sinh^{-1}(sinθ) = log(cot(θ/2)), using trigonometric identities and logarithmic transformations. Suitable for students studying advanced trigonometry and hyperbolic functions.

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