Question

write the function f(x) = cosh(2x)+2sinh^2x in exponential form

Answer 1

\[
\cosh ( 2 x) = \frac { e^{2x} + e^{-2x}}2\\
\sinh ( 2 x) = \frac { e^{2x} - e^{-2x}}2\\
\]

Answer 2

Add them up. It should be easy to find the answer.

Answer 3

Its sinh^2(x) its square not 2x

Answer 4

apply the same concept as shown then.

Answer 5

\[ \cosh ^2( x) = \left ( \frac { e^{x} + e^{-x}}2 \right)^2\\
\sinh ^2 x = \left ( \frac { e^{x} - e^{-x}}2\right)^2\\ \]

Answer 6

Does this mean that cos and sin functions are actually two functions combined?

Answer 7

\[
\\sinh^2(x) =\frac {e^{4x} -2 + e^{-4x}}{4}

\]

Answer 8

\[
2 \sinh^2(x) =\frac {e^{4x} -2 + e^{-4x}}{2}
\]

Answer 9

Now add cosh(2x) to the above and you will be done.

Answer 10

Recall that
\[\cosh ( 2 x) = \frac { e^{2x} + e^{-2x}}2\\
\]

Answer 11

Add
\[
\cosh ( 2 x) = \frac { e^{2x} + e^{-2x}}2\\
\text { to }\\
2 \sinh^2(x) =\frac {e^{4x} -2 + e^{-4x}}{2}
\]
and you are done.