In the sentence "I taught my cat some tricks", the direct object is "tricks" and the indirect object is "cat".
In the sentence "I taught my cat", what are the direct and indirect objects, if any?
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(1) "I taught my cat some tricks."
(2) "I taught my cat."
I must disagree with @user111737. The mere fact that the speaker no longer specifies what he is teaching the cat does not convert the cat from the indirect object in sentence (1) to the direct object in sentence (2). In either sentence, the cat is the recipient of what is being taught. Of course, I never realized that cats could be taught anything. ;-)
Consider the sentence
S V I D
Where "S" is the subject, "V" is the verb, "I" is the indirect object, and "D" is the direct object.
An example of that would be
I [S] gave [V] him [I] the book [D]
The sentence can be changed from active to passive in the following ways
D was V to I by S
Where "D" becomes the subject of the passive sentence, and "I" and "S" become objects of preposition. My example would become
The book [D] was given [V] to him [I] by me [S]
Another way to make the sentence passive is like this:
I was V D by S
Where "I" becomes the subject of the passive sentence, "D" stays the direct object, and "S" becomes an object of preposition. The example would be:
He was given the book by me
When a sentence such as the first one you gave is in active form, it can be hard to see where the indirect object is. But if you find the two passive versions of the sentence, then the indirect object becomes obvious.
Your original sentence was
I taught my cat some tricks 
It can be changed to a passive sentence two ways:
My cat was taught some tricks my me 
Some tricks were taught to my cat by me 
In sentence , one might be tempted to point out that "cat" is the subject of that passive sentence and thus the direct object of sentence . However, sentence  is simply of the form "I was V D by S" which I showed you before. On the other hand, sentence  is of the form "D was V to I by S" (as is evident by the words "to" and "by"). Both sentences suggest that "cat" is the indirect object.
Therefore, in your first sentence, "I" is the subject, "taught" is the verb, "cat" is the indirect object, and "tricks" is the direct object.
Now consider the second sentence
I taught my cat
What is the action? "Taught" Who or what performed the action? "I" What or whom did the subject teach? "Cat"
There is no indirect object, but only a direct object. There is only one way to make this sentence passive:
My cat was taught by me
Now it becomes obvious that "cat" is the direct object of the first sentence. In a passive sentence with no direct object, the sentence is the direct object of an active version of the sentence.
Furthermore, you cannot precede "cat" by "to" ("I taught to my cat") indicating that it is indeed not an indirect object. On the other hand, you could precede "cat" with "to" in the first sentence you gave ("I taught some tricks to my cat") but not "tricks" ("I taught my cat to some tricks")
One more thing, a sentence cannot have an indirect object without a sirect object. See the sources below*
In conclusion, "cat" is the indirect object and "tricks" is the direct object of the first sentence you gave, and "cat" is a direct object in the second sentence you gave
*Sources that you can't have an indirect object without a direct object: