Yes, you need to apply patch on both primary and standby.
From oracle documentation:
You should use the same version, release, and patch of the Oracle RDBMS for the primary and standby databases so that failover operations are not compromised.
It is recommended to apply the patch on both primary and standby database. Standby database is a mirrored copy of the primary database and Oracle recommends to have the patches to be same on both primary and standby database.
Not sure why you are getting Oracle 8 links on this question.
MOS doc :
How do you apply a Patchset,PSU or CPU in a Data Guard Physical Standby configuration [ID 278641.1]
In a nutshell it this :
1. Disable log shipping from the Primary
2. Shutdown Standby
3. Install the CPU patch software on Standby server
4. Startup Standby in recovery mode (do NOT run any SQL the standby)
5. Shutdown Primary
6. Install the CPU patch software on Primary
7. Run the SQL for the patch on Primary
8. Re-enable log shipping
9. Monitor the redo apply from Primary to Standby --- this will also upgrade the Standby data dictionary