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.