Which mantras should be recited to properly worship Lord Rudra?
Although it's been said that Lord Shiva is very easily pleased but the method of his worship is by no way simple or short.
It's easily the most elaborate process at least if you want to do it as the scriptures has ordained.
You can see a lot of mantras in Tezz's answer but the point is that not everyone is allowed to chant a lot of those mantras including in particular the Veda mantras.
So,i don't think you will benefit a lot by knowing the details of a ritualistic Shiva Puja or even the mantras used there in because that's a huge subject in itself.Instead,i think, you should always approach Shiva through devotion.
Parama Shiva himself recommends that path in Kurma Purana.
So basically i'm answering partially your question with my focus being on this question : "What is the proper way to worship Lord Rudra" ?
Parama Shiva asserted:
Naaham Tapobhirvividhairna Daanena na cheyjyayaa, Shakyo hi Purushair jaaturmutey Bhaktiamanuttamaam||
I am always realizable by Bhakti or Pure Devotion and not necessarily by Tapasya, Daana or Charity and Yagnaas!
Na madbhaktaa vinashyanti madbhaktaa veeta kalmashaah, Aadaavetat pratigjaanam na mey bhaktah pranashyati|| Patram Pushpam Phalam toyam madaaraadhana kaaranaat, Yo mey dadaati niyatah sa meyBhaktah Priyomatah||
My Bhaktaas would never be destroyed nor harmed but would be washed off their sins as I took a vow that my Bhaktas are protected. Bhaktas are merely required to perform my Puja with utmost sincerity and offer leaves, flowers, fruits and even water as I would be pleased.
So,you are merely required to worship him with water,fruits and leaves as per your capabilities.Lord Shiva himself does not emphasize on a "proper way".Because he knows very well that not everyone can worship him with all the needed mantras from Vedas and other Shastras.That is even very difficult for well-versed Vedic Purohits so what to say of the common man.
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