Prove the identity
tan2(x) - sin2(x) = tan2(x) sin2(x)
To prove the given identity, we need to manipulate the left side of the equation until it matches the right side. We will start with the left side and simplify it step by step.
1. Recall the identity: tan(2x) = 2tan(x) / (1 - tan^2(x)).
Applying this identity, we can rewrite the left side of the equation as:
tan^2(x) - sin^2(x) = [tan(2x) / 1] - sin^2(x)
2. Using the identity for sin^2(x) = 1 - cos^2(x), we can substitute it in the equation:
tan^2(x) - sin^2(x) = [tan(2x) / 1] - (1 - cos^2(x))
= tan(2x) - 1 + cos^2(x)
3. Next, we will express tan(2x) in terms of sin(2x) and cos(2x) using the identity:
tan(2x) = (2sin(2x)) / (1 - cos(2x))
4. Substituting this into the equation:
tan^2(x) - sin^2(x) = [(2sin(2x)) / (1 - cos(2x))] - 1 + cos^2(x)
5. Let's simplify the equation further:
tan^2(x) - sin^2(x) = (2sin(2x) - 1 + cos^2(x)(1 - cos(2x))) / (1 - cos(2x))
6. Rearranging the terms, we get:
tan^2(x) - sin^2(x) = (2sin(2x) - 1) / (1 - cos(2x)) + (cos^2(x)(1 - cos(2x))) / (1 - cos(2x))
7. Now, let's simplify the fractions:
tan^2(x) - sin^2(x) = (2sin(2x) - 1) / (1 - cos(2x)) + cos^2(x)
8. Expanding cos^2(x) gives:
tan^2(x) - sin^2(x) = (2sin(2x) - 1) / (1 - cos(2x)) + cos^2(x)(1 - cos(2x)) / (1 - cos(2x))
9. Simplifying further:
tan^2(x) - sin^2(x) = (2sin(2x) - 1 + cos^2(x)(1 - cos(2x))) / (1 - cos(2x))
10. Finally, we can observe that the right side of the equation matches the expression in step 3:
tan^2(x) - sin^2(x) = tan^2(x)sin^2(x)
Hence, we have proven the given identity:
tan^2(x) - sin^2(x) = tan^2(x)sin^2(x).