Rcosalpha, how do I know to use double angle formula and not the factor formula?

For example:

Solve between 0 and 360, 8cosθ + 15sinθ = 10

How do I know to use the double angle formula and not the factor formula?

Also, I've seen the solutions so I know it's the double angle formula. However, none of the formulas (eg. Cos(A+-B) = CosACosB -+ sinAsinB) have just a cos and a sin in them, they have a coscos and a sinsin (just like the one I just said as an example), so how do you know which of the double angle formulas to use?

Thanks

I would go with the latter, actually. Consider

$R\sin(\theta + \alpha) = R\sin(a)\cos\theta + R\cos(a)\sin\theta$.

You can compare the terms,

$R\sin(a)\cos\theta = 8\cos\theta \implies R\sin(a) = 8$

and

$R\cos(a)\sin\theta = 15\sin\theta \implies R\cos(a) = 15$ .

Are you familiar with determining $R$ and $\alpha$ here?

What do you mean by you would go with the latter?

Yes, I do know how to compare terms to determine R and alpha. Looking at the mark scheme, I don't see how your way would work since tan alpha = Rsin(a)/Rcos(a) which would be 8/15, like the mark scheme, rather than your way which would give tan alpha to be 15/8, which would mess up the value for alpha wouldn't it?