I'll write x for theta.
If we expand along the third row, the determinant is
-cos(x)+2cos(x)(2(cos(x))^2-1)=4(cos(x))^3-3cos(x)=cos(x)(4(cos(x))^2-3)=cos(x)(2(2(cos(x))^2-1)-1)=cos(x)(2cos(2x)-1)=2cos(2x)cos(x)-cos(x)=cos(2x+x)=cos(3x)
I want to solve this problem by using the properties of determinants.
The I suggest you do elementary row operations to get the determinant into upper triangular form and then apply the appropriate theorems to calculate the determinant.