Tan@ + tan2@ + tan3@ ... tann@ = ?
The expression you have provided represents the sum of tangent values of angles @, 2@, 3@, and so on, up to n@.
To find a general formula for the sum, let's consider a simpler case where we look at the sum of tangent values of consecutive angles:
tan@ + tan2@ + tan3@ + ... + tan(n-1)@.
We can use the tangent addition formula to rewrite each term in terms of the sum of the previous terms:
tan2@ = (tan@ + tan@) / (1 - tan@ * tan@),
tan3@ = (tan2@ + tan@) / (1 - tan2@ * tan@),
tan4@ = (tan3@ + tan@) / (1 - tan3@ * tan@),
And so on until the (n-1)-th term:
tan(n-1)@ = (tan(n-2)@ + tan@) / (1 - tan(n-2)@ * tan@).
We can recursively apply this process to simplify the sum until we reach the first term:
tan(n-2)@ = (tan(n-3)@ + tan@) / (1 - tan(n-3)@ * tan@),
...
tan2@ = (tan@ + tan@) / (1 - tan@ * tan@).
Substituting back into the original sum:
tan@ + [(tan@ + tan@) / (1 - tan@ * tan@)] + [(tan@ + tan2@) / (1 - tan2@ * tan@)] + ... + [(tan(n-2)@ + tan@) / (1 - tan(n-2)@ * tan@)].
If you simplify this expression, you will notice that many terms cancel out until you are left with just the first and last terms:
tan@ + [(tan@ + tan@) / (1 - tan@ * tan@)] + [(tan@ + tan2@) / (1 - tan2@ * tan@)] + ... + [(tan@ + tan(n-1)@) / (1 - tan(n-1)@ * tan@)].
The numerator of each term contains two tangent values, and the denominator contains the product of those two tangent values. If we group the terms with respect to the common factor of the tangent value @, we get:
[tan@ * (1 + 1/(1 - tan@ * tan@))] + [tan@ * (1 + 1/(1 - tan2@ * tan@))] + ... + [tan@ * (1 + 1/(1 - tan(n-1)@ * tan@))].
Notice that the term within the parentheses is always the same, regardless of the index of summation. Let's denote it as K:
K = 1 + 1/(1 - tan@ * tan@).
Now we can rewrite the sum in a more compact form:
tan@ * (K + K + ... + K),
tan@ * (n * K).
Therefore, the sum of all the tangent values starting from @ up to n@ is equal to:
tan@ * n * (1 + 1/(1 - tan@ * tan@)).
This formula can be used to find the sum of tangent values for any given values of @ and n. Just substitute the specific angle values into the formula above.