How do you proove the identity

tanxsin2x = 2-2cos^2x

tanx sin2x=2sinxcosx(sinx/cosx)=

2sin^2 x= 2*(1-cos^2 x)

To prove the identity tan x sin 2x = 2 - 2cos^2 x, we need to simplify the left-hand side (LHS) and show that it is equal to the right-hand side (RHS). Here's how you can do it step by step:

1. Start with the LHS and use the double-angle formula for sine: sin 2x = 2sin x cos x. This gives us: tan x * 2sin x cos x.

2. Next, express tan x as sin x / cos x: (sin x / cos x) * 2sin x cos x.

3. Simplify the expression by canceling out the common factors: 2sin^2 x.

4. Now, let's simplify the RHS: 2 - 2cos^2 x.

5. We can factor out a 2 from both terms: 2(1 - cos^2 x).

6. Recognize that 1 - cos^2 x is the identity for sin^2 x: 2(sin^2 x).

7. Finally, we have shown that the LHS equals the RHS: 2sin^2 x = 2sin^2 x.

Therefore, we have proven the identity tan x sin 2x = 2 - 2cos^2 x.