OCR A Level: Further Pure 2 - Hyperbolics [January 2010 Q5]

$(\text{i})$ Using the definitions of $\sinh x$ and $\cosh x$ in terms of $e^x$ and $e^{-x}$ , show that $\cosh^2 x - \sinh^2 x \equiv 1 .$

$(\text{ii})$ Solve the equation $2 \tanh^2 x - \text{sech} \, x = 1$ giving your answer(s) in logarithmic form.

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If the sum of all the solutions is $\ln a$ , input $a$ as your answer.

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There are 4 marks available for part (i) and 4 marks for part (ii).

In total, this question is worth 11.1% of all available marks in the paper.

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This is part of the set OCR A Level Problems .