Simple Harmonic Oscillator Squeezing

[t0pquark]

Homework Statement

How to derive $(\hat{a}cosh\gamma + \hat{a^\dagger}sinh\gamma \vert 0_{\gamma} \rangle = 0$ from $\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger} \vert 0 \rangle$

Relevant Equations

Creation operator: $\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma$
Annihilation operator: $\hat{a}_2^\dagger = -\hat{a}_1sinh\gamma + \hat{a}_1^\dagger cosh\gamma$
Where $e^\gamma \equiv \sqrt{\frac{m_1 \omega_1}{m_2 \omega_2}}$

I'm working through https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_06.pdf, and I'm stumped how they got from Equation 5.26 ($\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger} \vert 0 \rangle$) to Equation 5.27 ($(\hat{a}cosh\gamma + \hat{a^\dagger}sinh\gamma) \vert 0_{\gamma} \rangle = 0$).
I've tried substituting in the creation ($\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma$) and annihilation (($\hat{a}_2^\dagger = -\hat{a}_1 sinh\gamma + \hat{a}_1^\dagger cosh\gamma$) operators and then rearranging, but I can't get what they have.

 

[tnich]

Equation 5.27 does not follow from 5.26. It is a repetition of 5.12.