Which of the following expression is always true?

Which of the following expression is always true?
  1. $\cosh \frac{x}{2}=\sqrt{\frac{\cosh x-1}{2}}$
  2. $\tanh \left(\frac{x}{2}\right)=\frac{2 \sinh x}{\cosh x+1}$
  3. $\sinh \left(\frac{x}{2}\right)=\frac{\sinh x}{\sqrt{2(\cosh x+1)}}$
  4. $\sinh \left(\frac{x}{2}\right)=\frac{\sinh x}{\sqrt{2(\cosh x-1)}}$

Solution

(a) $\cosh \frac{x}{2}=\sqrt{\frac{1+\cosh x}{2}}$ (b) $\tanh \left(\frac{x}{2}\right)=\frac{\sinh x}{\cosh x+1}$ (c) $\sinh \left(\frac{x}{2}\right)=\frac{\sinh x}{\sqrt{2(\cosh x+1)}}$

Asked in: AP EAMCET 2021 (25 Aug Shift 1)

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