If f( )= ₁ ₂ ₃ _n , then ₁+ ₂+ ₃+ + _ n =

Step-by-step solution

The product of cosines is defined as f( )= ₁ ₂ ₃ _n . The sum of tangents ₁+ ₂+ ₃+ + _n must be expressed in terms of f( ) and its derivative. Taking the natural logarithm of f( ) utilizes the logarithmic product identity: f( )= _ i=1 ^ n ( _i) . Differentiation with respect to is applied, assuming d _i d =1 as standard in such problems. The left side is f'( ) f( ) , and the right side becomes _ i=1 ^ n 1 _i (- _i) . Since - _i _i = - _i , the derivative equation simplifies to f'( ) f( ) = - _ i=1 ^ n _i . Solving