Question

Which of the following is an identity?

A. sinx - cosx + 1 = tanx
B. (1 - 2sin^2x)csc^2 x = 4cos2x - 2
C. sin^2xcot^2x + cos^2xtan^2x = 1
D. tan^2x + cot^2x = 1

Answer 1

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

`cot(2x) = (cos(2x))/(sin(2x)) \\ tan(2x) = (sin(2x))/(cos(2x)) \\ cos^(2)(2x)+sin^(2)(2x) = 1`

Therefore

`sin^(2)(2x) \, cot^(2)(2x)+cos^(2)(2x) \, tan^(2)(2x) \\ =sin^(2)(2x) (cos^(2)(2x))/(sin^(2)(2x)) +cos^(2)(2x) (sin^(2)(2x))/(cos^(2)(2x)) \\ =cos^(2)(2x) + sin^(2)(2x) \\ =1`
This proves the identity.

Answer: C