P1775 is the most relevant code. It means Solenoid valve latched in torque convertor.
P0700 is consequence of the P1775 code. P0700 means that there is an additional code stored in the TCM (transmission control module). The TCM stored code is probably related to the same problem. TCM codes can be pulled out at the car dealer or at an authorized garage. On this see: I get code p0700 after an OBDII scan.
P1775 is often coming with misfire codes.
P1775 means that the TCM detects that the solenoid valve located in the valve body (inside transmission pan under the filter) is not in the expected position (valve has two positions LR and TCC).
This cause rough shifting and driving problems.
Sometimes it is possible fixing the problem by resetting TCM and ECU. This is done disconneting battery cable (ensure you have anti-theft code if needed).
However, if a P1775 occurred, that means that there had been a solenoid valve malfunction, and problem must be fixed.
The first thing to do is replacing transmission fluid, and if needed also the filter. If the fluid level is poor, or if fluid is dark brown and satured with metal particles, that can cause solenoid valve malfunction.
If replacing the fluid does not help, then the most common problem causing the P1775 code is a defective solenoid in the solenoid pack.
Transmission shift solenoids are located in valve body, and they can be tested reading Ohms with a multimeter.
If that does not fix the problem (usually it does), the mechanic will proceed testing wiring to solenoid pack, TCM and solenoid valve, to find the reason of malfunction. Most common problems causing P1775 are related to tranny fluid and solenoid pack.
On automatic transmission problems see also: My car does not shift..