Proving that two determinants are equal without expanding them

Question

So I need to prove that

$$ \begin{vmatrix} \sin^2(\alpha) & \cos(2\alpha) & \cos^2(\alpha) \\ \sin^2(\beta) & \cos(2\beta) & \cos^2(\beta) \\ \sin^2(\gamma) & \cos(2\gamma) & \cos^2(\gamma) \\ \end{vmatrix} $$ $$ = \begin{vmatrix} \sin(\alpha) & \cos(\alpha) & \sin(\alpha + \delta) \\ \sin(\beta) & \cos(\beta) & \sin(\beta + \delta) \\ \sin(\gamma) & \cos(\gamma) & \sin(\gamma + \delta) \\ \end{vmatrix} $$

Now, $$ \begin{vmatrix} \sin^2(\alpha) & \cos(2\alpha) & \cos^2(\alpha) \\ \sin^2(\beta) & \cos(2\beta) & \cos^2(\beta) \\ \sin^2(\gamma) & \cos(2\gamma) & \cos^2(\gamma) \\ \end{vmatrix} = \begin{vmatrix} \sin^2(\alpha) & \cos^2(\alpha) - \sin^2(\alpha) & \cos^2(\alpha) \\ \sin^2(\beta) & \cos^2(\beta) - \sin^2(\beta) & \cos^2(\beta) \\ \sin^2(\gamma) & \cos^2(\gamma) - \sin^2(\gamma) & \cos^2(\gamma) \\ \end{vmatrix} $$

Adding column $1$ to column $2$ then makes column $2$ and column $3$ equal and hence the first determinant $ = 0$.

I'm stuck in trying to prove that the second determinant is also zero, so I need help in that.

Answer 1

$$\begin{vmatrix} \sin(\alpha) & \cos(\alpha) & \sin(\alpha + \delta) \\ \sin(\beta) & \cos(\beta) & \sin(\beta + \delta) \\ \sin(\gamma) & \cos(\gamma) & \sin(\gamma + \delta) \\ \end{vmatrix}=\begin{vmatrix} \sin(\alpha) & \cos(\alpha) & \sin\alpha\color{red}{\cos\delta}+\color{red}{\sin\delta}\cos\alpha \\ \sin(\beta) & \cos(\beta) & \sin\beta\color{red}{\cos\delta}+\color{red}{\sin\delta}\cos\beta \\ \sin(\gamma) & \cos(\gamma) & \sin\gamma\color{red}{\cos\delta}+\color{red}{\sin\delta}\cos\gamma \\ \end{vmatrix}$$$${}$$

$$=\begin{vmatrix} \sin(\alpha) & \cos(\alpha) & \sin\alpha \\ \sin(\beta) & \cos(\beta) & \sin\beta\\ \sin(\gamma) & \cos(\gamma) & \sin\gamma \\ \end{vmatrix}\color{red}{\cos\delta}+\begin{vmatrix} \sin(\alpha) & \cos(\alpha) & \cos\alpha \\ \sin(\beta) & \cos(\beta) & \cos\beta\\ \sin(\gamma) & \cos(\gamma) & \cos\gamma \\ \end{vmatrix}\color{red}{\sin\delta}=0+0=0$$

Answer 2

HINT: Relate the third column to the first two using the trigonometric identity $$\sin(\theta+\delta)=\sin\theta\cos\delta+\cos\theta\sin\delta.$$