Further hyperbolic identities and proofs Quiz

1. Given $\tanh x=\frac45$, find $\operatorname{sech}^2x$.

• A. $\frac{3}{5}$
• B. $\frac{16}{25}$
• C. $\frac{9}{25}$
• D. $\frac{41}{25}$

2. Given $\sinh x=\frac{5}{12}$ and $x>0$, find $\sinh 2x$.

• A. $\frac{65}{72}$
• B. $\frac{5}{6}$
• C. $\frac{25}{144}$
• D. $\frac{97}{72}$

3. Which identity is correct?

• A. $\operatorname{cosech}^2x=1-\coth^2x$
• B. $\operatorname{sech}^2x=\tanh^2x-1$
• C. $\cosh 2x=\cosh^2x-\sinh^2x$
• D. $\operatorname{cosech}^2x=\coth^2x-1$

4. Given $\cosh 2x=5$ and $x>0$, find $\sinh x$.

• A. $\sqrt2$
• B. $2$
• C. $\sqrt3$
• D. $\frac{1}{\sqrt2}$

5. Using $\tanh 2x=\frac{2t}{1+t^2}$ with $\tanh 2x=\frac35$ gives $3t^2-10t+3=0$. Which is a valid value of $\tanh x$?

• A. $3$
• B. $\frac13$
• C. Both $3$ and $\frac13$
• D. Neither