I have the following problem. I have two vectors $ \begin{bmatrix} \alpha_1\\ \beta_1 \end{bmatrix} $ and $ \begin{bmatrix} \alpha_2\\ \beta_2 \end{bmatrix} $ where the entries of the vector are complex numbers and the length of both vectors is 1. I also have the following (bigger vector): $$ V = \frac{1}{2}\begin{bmatrix} 2\alpha_1\alpha_2\\ \alpha_1\beta_2 + \alpha_2\beta_1 \\ \alpha_1\beta_2 + \alpha_2\beta_1\\ 2\beta_1\beta_2 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} $$ Now I calculate V$^\dagger$V where V$^\dagger$ is the complex conjugate of V. This gives:
$$ V^\dagger V = \begin{bmatrix} \alpha_1^*\alpha_2^* & \frac{\alpha_1^*\beta_2^* + \alpha_2^*\beta_1^*}{2} & \frac{\alpha_1^*\beta_2^* + \alpha_2^*\beta_1^*}{2} & \beta_1^*\beta_2^* & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \alpha_1\alpha_2\\ \frac{\alpha_1\beta_2 + \alpha_2\beta_1}{2} \\ \frac{\alpha_1\beta_2 + \alpha_2\beta_1}{2}\\ \beta_1\beta_2 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} $$ $$ = \alpha_1^*\alpha_1\alpha_2^*\alpha_2 + \frac{(\alpha_1^*\beta_2^* + \alpha_2^*\beta_1^*)(\alpha_1\beta_2 + \alpha_2\beta_1)}{4} + \beta_1^*\beta_1\beta_2^*\beta_2 $$
Where the star denotes the complex conjugate of the number. Now I have to prove that the number above is equal to the following:
$$ V^\dagger V = \frac{1}{2} + \frac{(\alpha_1^*\alpha_2+\beta_1^*\beta_2)(\alpha_2^*\alpha_1+\beta_2^*\beta_1)}{2} $$
I tried rewriting and working out the term above but that didn't work and I probably made a mistake somewhere. Also, after writing out the whole equation, I was not sure how to proceed. I am wondering if there is an easy way to show the validity of this equation. Any help would be greatly appreciated!