Of the given expressions, expression C seems the most likely to be an identity.
To prove it, we should show that
sin²(x) cot²(2x) + cos²(2x) tan²(2x) = 1
Note that
[tex]cot(2x) = \frac{cos(2x)}{sin(2x)} \\ tan(2x) = \frac{sin(2x)}{cos(2x)} \\ cos^{2}(2x)+sin^{2}(2x) = 1[/tex]
Therefore
[tex]sin^{2}(2x) \, cot^{2}(2x)+cos^{2}(2x) \, tan^{2}(2x) \\ =sin^{2}(2x) \frac{cos^{2}(2x)}{sin^{2}(2x)} +cos^{2}(2x) \frac{sin^{2}(2x)}{cos^{2}(2x)} \\ =cos^{2}(2x) + sin^{2}(2x) \\ =1[/tex]
This proves the identity.
Answer: C
