#include <iostream>
#include <cstdlib>
#include <string>
using namespace std;

string S, A;
unsigned short An = 1;
bool negative     = true;

void get_next_A();
void get_S(unsigned short);

int main()
{
    unsigned short input_n;

    cin >> input_n;

    get_S(input_n);

    cout << S << endl;

    return 0;
}

void get_next_A()
{
    if(An == 1)
        A = "sin(1)";
    else
    {
        A.erase(A.size() - An + 1, An);

        if(negative)
            A += "-sin(";
        else
            A += "+sin(";

        char buffer[5];

        itoa(An, buffer, 10);
        A += buffer;

        for(unsigned short j = 0; j < An; ++j)
            A += ")";

        negative = !negative;

    }

    ++An;
    return;
}

void get_S(unsigned short n)
{
    char _buffer[10];

    for(unsigned short i = 0; i < n - 1; ++i)
        S += '(';

    for(unsigned short i = n; i > 0; --i)
    {
        get_next_A();

        itoa(i, _buffer, 10);

        S += A;
        S += '+';
        S += _buffer;
        if(i != 1)
            S += ')';

    }

    return;
}

You can make it simpler :)
For example, here is the function generating sine expressions:
string get_sine_expression(int x, int n)
{
  if (x == n) return "sin(" + to_string(x) + ")";

  char sign = x & 1 ? '-' : '+';

  return "sin(" + to_string(x) + sign + get_sine_expression(x + 1, n) + ")";
}