C :

#include <stdio.h>
#define N 200000

int main()
{
    char* sinA = "sin(%d%s%s)";
    char* Sa = "%s+%d";
    char* S = "(%s)%s";
    char An[N] = "sin(1%s%s)", An2[N], An3[N], Sn[N] = "sin(1)+%d", Sn2[N];
    int i, j, n;
    scanf("%d", &n);
    sprintf(Sn2, Sn, n);
    strcpy(Sn, Sn2);
    for (i = 2; i <= n; i++)
    {
        sprintf(An2, An, i % 2 == 0 ? "-" : "+", sinA);
        strcpy(An, An2);
        sprintf(An2, An, i, "%s", "%s");
        strcpy(An, An2);
        sprintf(An2, Sa, An, n - i + 1);
        sprintf(An3, An2, "", "");
        sprintf(Sn2, S, Sn, An3);
        strcpy(Sn, Sn2);
        sprintf(Sn2, Sn, n - i+ 1);
        strcpy(Sn, Sn2);
    }
    printf("%s", Sn);
    return 0;
}

C++ :

#include <stdio.h>
#include <string.h>
#define N 200000

int main()
{
    char* sinA = "sin(%d%s%s)";
    char* Sa = "%s+%d";
    char* S = "(%s)%s";
    char An[N] = "sin(1%s%s)", An2[N], An3[N], Sn[N] = "sin(1)+%d", Sn2[N];
    int i, j, n;
    scanf("%d", &n);
    sprintf(Sn2, Sn, n);
    strcpy(Sn, Sn2);
    for (i = 2; i <= n; i++)
    {
        sprintf(An2, An, i % 2 == 0 ? "-" : "+", sinA);
        strcpy(An, An2);
        sprintf(An2, An, i, "%s", "%s");
        strcpy(An, An2);
        sprintf(An2, Sa, An, n - i + 1);
        sprintf(An3, An2, "", "");
        sprintf(Sn2, S, Sn, An3);
        strcpy(Sn, Sn2);
        sprintf(Sn2, Sn, n - i+ 1);
        strcpy(Sn, Sn2);
    }
    printf("%s", Sn);
    return 0;
}
//((sing(1)+3)sin(1-sin(2)+2)sin(1-sin(2+sin(3))))+1

Java :

import java.util.Scanner;

public class Main {
	public static void main(String[] args) {
		Scanner in = new Scanner(System.in);
		int n = in.nextInt();
		for(int i=1;i<n;i++){
			System.out.print("(");
		}
		String s = f2(1,n);
		System.out.print(s.substring(0, s.length()-1));
		

	}
	public static String f(int i,int n){
		if(n==1) return "sin(1)";
		if(i-1==n) return "";
		if(f(i+1,n).equals("")){
			if(i%2==1) return "sin("+i+f(i+1,n)+")";
			else return "sin("+i+f(i+1,n)+")";
		}
		if(i%2==1) return "sin("+i+"-"+f(i+1,n)+")";
		else return "sin("+i+"+"+f(i+1,n)+")";
	}
	public static String f2(int i,int n){
		if(n==0) return "";
		if(i%2==1) return f(1,i)+"+"+n+")"+f2(i+1,n-1);
		else return f(1,i)+"+"+n+")"+f2(i+1,n-1);
	}
}