Forces on Pendulum: Resolving mg & T

[midlifephy]

I find that the forces acting on a pendulum bob are treated in two different (contradictory?) ways.
Let the angle that the cord makes with the vertical be x, the mass of the bob m and the tension in the cord T.
1. In one case, the weight of the bob is resolved into components so that mgcos(x) is along T, so T = mgcos(x).
2. In another case, the tension is resolved into ITS components, so that mg = Tcos(x).
Obviously only one of them can be correct at one time. I can't figure out which of them to use and when.
I am a physics teacher, by the way.
Thanks in advance.

 

[bp_psy]

1. In one case, the weight of the bob is resolved into components so that mgcos(x) is along T, so T = mgcos(x).

Why do you think that statement is true?
"mgcos(x) is along T" does not imply "T = mgcos(x)"
Draw a free body diagram.

 

[K^2]

I find that the forces acting on a pendulum bob are treated in two different (contradictory?) ways.
Let the angle that the cord makes with the vertical be x, the mass of the bob m and the tension in the cord T.
1. In one case, the weight of the bob is resolved into components so that mgcos(x) is along T, so T = mgcos(x).
2. In another case, the tension is resolved into ITS components, so that mg = Tcos(x).
Obviously only one of them can be correct at one time. I can't figure out which of them to use and when.
I am a physics teacher, by the way.
Thanks in advance.

Neither of these two are correct.

ma_r = mgcos(x) - T
a_r = -L\dot{x}^2