prove:

tan*2x-sin*2x=sin*2xtan*2x

To prove the given statement: tan(2x) - sin(2x) = sin(2x)tan(2x), we'll use the basic trigonometric identities and algebraic manipulation.

1. Start with the left side of the equation: tan(2x) - sin(2x).
2. Apply the double-angle formula for tangent: tan(2x) = (2tan(x))/(1-tan^2(x)).
3. Use the double-angle formula for sine: sin(2x) = 2sin(x)cos(x).
4. Substitute the formulas from step 2 and step 3 into the equation:
(2tan(x))/(1-tan^2(x)) - 2sin(x)cos(x).
5. Combine the terms by finding a common denominator:
(2tan(x) - 2sin(x)cos(x) + 2sin(x)cos(x)tan^2(x))/(1-tan^2(x)).
6. Cancel out the terms:
(2tan(x))/(1-tan^2(x)).
7. Simplify further:
(2tan(x))/[(1 - tan(x))(1 + tan(x))].
8. Use the identity tan^2(x) + 1 = sec^2(x) to further simplify:
(2tan(x))/(sec^2(x) - tan^2(x)).
9. Apply the identity sec^2(x) - tan^2(x) = 1:
(2tan(x))/(1) = 2tan(x).
10. Therefore, the left side of the equation is equal to 2tan(x).
11. The right side of the equation is sin(2x)tan(2x).
12. Apply the double-angle formula for sine: sin(2x) = 2sin(x)cos(x).
13. Substitute the formula from step 12 into the equation:
2sin(x)cos(x)tan(2x).
14. Apply the double-angle formula for tangent: tan(2x) = (2tan(x))/(1-tan^2(x)).
15. Substitute the formula from step 14 into the equation:
2sin(x)cos(x)[(2tan(x))/(1-tan^2(x))].
16. Use algebraic manipulation to simplify:
2tan(x)sin(x)cos(x)/[(1-tan^2(x))].
17. Cancel out terms: 2tan(x).
18. Therefore, the right side of the equation is also equal to 2tan(x).
19. Since the left side of the equation is equal to the right side, the given statement is proven to be true.

Overall, by applying the fundamental trigonometric identities and algebraic simplification, we showed that tan(2x) - sin(2x) is indeed equal to sin(2x)tan(2x).