If both sides of Trial Balance are equal, Does it mean that books of accounts are completely correct?

No.
In my experience I checked many accountants TB were equal but there books of accounts were not correct, many mall practice I observed during my audit and the financial reports were totally different with the reports made by old TB.

No it only means that the debits equal credits.
But it's still possible that there is an error in the books.
For example, a transaction may have been recorded in the wrong accounts.

No.
Because there are certain errors due to which trial balance does not change.
Some examples are as fallows error of omission error of commission error of principle compensating error etc....
however there are some errors which if made then they will definitely imbalance some of which are: Single Sided entry Debit and credit entries have been made but with different values; Two entries have been made but on same side etc....

A trial balance only checks the sum of debits against the sum of credits.
That is why it does not guarantee that there are no errors.
The following are the main classes of error that are not detected by the trial balance.
An error of original entry is when both sides of a transaction include the wrong amount.
For example, if a purchase invoice for £21 is entered as £12, this will result in an incorrect debit entry (to purchases), and an incorrect credit entry (to the relevant creditor account), both for £9 less, so the total of both columns will be £9 less, and will thus balance.
An error of omission is when a transaction is completely omitted from the accounting records.
As the debits and credits for the transaction would balance, omitting it would still leave the totals balanced.
A variation of this error is omitting one of the ledger account totals from the trial balance.
An error of reversal is when entries are made to the correct amount, but with debits instead of credits, and vice versa.
For example, if a cash sale for £100 is debited to the Sales account, and credited to the Cash account.
Such an error will not affect the totals.
An error of commission is when the entries are made at the correct amount, and the appropriate side (debit or credit), but one or more entries are made to the wrong account of the correct type.
For example, if fuel costs are incorrectly debited to the postage account (both expense accounts).
This will not affect the totals.
An error of principle is when the entries are made to the correct amount, and the appropriate side (debit or credit), as with an error of commission, but the wrong type of account is used.
For example, if fuel costs (an expense account), are debited to stock (an asset account).
This will not affect the totals.
Compensating errors are multiple unrelated errors that would individually lead to an imbalance, but together cancel each other out.
A Transposition Error is an error caused by switching the position of two adjacent digits.
Since the resulting error is always divisible by9, accountants use this fact to locate the mis-entered number.
For example, a total is off by72, dividing it by9 gives8 which indicates that one of the switched digits is either more, or less, by8 than the other digit.
Hence the error was caused by switching the digits8 and0 or1 and9.
This will also not affect the totals.

No: It does not mean that the books are completely correct.
The following errors could have been made:
1- Entering correct figures in the wrong account (but on the correct side)
2- Reversing entries so that both entries are made on the incorrect side of the each account
3- Entering the incorrect total on both sides of the account.

there may be some principle errors occur and error of ommission , error of commision etc..

Certainly not.
If you fudge accounting entry still trial balance will tally but that doesn't mean that your books of accounts are correct.

no.we get only arithmatical accuracy of book of account,ther might be many possibility to transactions error like errors of ommission ,participating errors ,capital errors etc,actually trial balnce is considering rough statement to ensure that all debit has equal credit

NO, you still need to check that entries are assigned to the proper accounts make all the adjustment entries.and check for any illogical balances.

its because following the double entry system of accounting.
each transaction is recorded in both places.
so if we pass an entry, there will be one account having debit balance and credit balance.