Looks like you have muscle spasm. Bed rest. Hot packs. Some pain gel for local applications. Some pain killer should help. Increase in pain with cough spasm and Radiation of pain to leg can be due to disc prolapse.
Apply Hot Fomentation twice daily.
Avoid bending in front.
Postural Correction- Sit Tall, Walk Tall.
Extension Exercises x 15 times x twice daily - lying on tummy, take left arm up for 3 seconds, then bring it down, right arm up for 3 seconds, bring down. Bring right leg up, hold for 3 seconds, bring it down. Then right leg up and hold for 3 seconds and bring it down. Repeat twice a day- 10 times. Bhujang Asana -- Lie flat on your stomach, keeping the palms out, bend the neck backward, take a deep breath and while holding it for 6 seconds, raise the chest up. Release breath and relax your body. Repeat the exercise 15 times twice daily.
Core Strengthening Exercises- Straight Leg Raised With Toes Turned Outward, repeat 10 times, twice a day.
Hello.......may be it is muscle spasm, if pain is crampy.. You need to take rest till it relieves.. Local massage is helpful... If it's paining badly you need to consult orthopedic surgeon.. Doctor will give muscle relaxing injection...cough is not related to back pain .... Because of increased intraabdominal pressure you feel so..