prove that if cotx(1+sinx)=4m and cotx(1-sinx)=4n, then (m^2-n^2)^2=mn
To prove the given statement, we can use the properties of trigonometric functions and algebraic manipulations. Here's an explanation of the steps:
1. Start with the expressions provided: cot(x)(1+sin(x)) = 4m and cot(x)(1-sin(x)) = 4n.
2. Use the identity cot(x) = cos(x)/sin(x), and rewrite the expressions: (cos(x)/sin(x))(1+sin(x)) = 4m and (cos(x)/sin(x))(1-sin(x)) = 4n.
3. Simplify the expressions: cos(x) + cos(x)sin(x)/sin(x) = 4m and cos(x) - cos(x)sin(x)/sin(x) = 4n.
4. Cancel out sin(x) in the numerators: cos(x) + cos(x) = 4m and cos(x) - cos(x) = 4n.
5. Combine the like terms: 2cos(x) = 4m and 0 = 4n.
6. Divide both sides of the equations by 2 and simplify: cos(x) = 2m and 0 = 4n.
7. From the second equation, we can conclude that n = 0, since any value multiplied by 4 and resulting in zero must be zero.
8. Substitute this value back into the first equation: cos(x) = 2m.
9. Square both sides of the equation: cos^2(x) = (2m)^2.
10. Use the identity cos^2(x) = 1 - sin^2(x): 1 - sin^2(x) = 4m^2.
11. Rearrange the equation: sin^2(x) = 1 - 4m^2.
12. Square both sides of the equation: sin^4(x) = (1 - 4m^2)^2.
13. From the given equations, we know that (1 - sin(x)^2) = cos^2(x). Therefore, cos^4(x) = (1 - 4m^2)^2.
14. Finally, since (cos^2(x))^2 = cos^4(x), we have (cos^2(x))^2 = (1 - 4m^2)^2.
15. Substituting back cos^2(x) = (2m)^2 gives us (2m^2)^2 = (1 - 4m^2)^2.
16. Simplifying both sides, we get (4m^4) = (1 - 4m^2)^2.
17. Expand the right side of the equation: (4m^4) = 1 - 8m^2 + 16m^4.
18. Rearrange the equation: 16m^4 - 4m^4 + 8m^2 - 1 = 0.
19. Combine like terms: 12m^4 + 8m^2 - 1 = 0.
20. Factor the equation: (3m^2-1)(4m^2+1) = 0.
21. Set each factor equal to zero: 3m^2-1 = 0 or 4m^2+1 = 0.
22. Solve each equation: 3m^2 = 1 or 4m^2 = -1.
23. The second equation has no real solutions, so we focus on the first equation: 3m^2 = 1.
24. Solve for m: m^2 = 1/3.
25. Take the square root to get two possible values for m: m = ±√(1/3).
26. From step 7, we know that n = 0.
27. Now, substitute the values of m and n into the expression (m^2 - n^2)^2 = mn.
28. (m^2 - n^2)^2 = (√(1/3))^2 - 0^2)^2 = (1/3) - 0)^2 = (1/3)^2 = 1/9.
29. mn = (√(1/3)) * 0 = 0.
Therefore, (m^2 - n^2)^2 is equal to 1/9 and mn is equal to 0. So, (m^2 - n^2)^2 is indeed equal to mn.