hello seanmac,
First of all, I need to alert you about what interval are we solving this on?
This is critical as for cos or sin are periodic function! So they really infinite number of solutions.
Now that aside,
This is how we solve this.
Begin by factoring cos(α) (I will use alpha instead don't bother we can change the variable to any letter it does not change the problem, I just don't have theta in my keyboard)
so this we get cos(α)[1-sin(α)]=0
i.e
cosα=0 or 1-sinα=0
cosα=0 or sinα=1
for cosα=0
this happen whenever α=kpi/2 for k an odd integer
the other equation sinα=1
gives α=kpi/2 for k an odd integer
since both equation gives exactly the same the solutions
then the solution is just α=kpi/2 for any odd number k