Which choice completes the identity? \( 1-\csc ^{2} x= \) \( \tan ^{2} x \) \( -\tan ^{2} x \) \( \cot ^{2} x \) \( -\cot ^{2} x \) \( 1-\sec ^{2} x \)

To complete the identity \( 1-\csc^{2} x= \), we can use the Pythagorean identities. We know that \( \csc^{2} x = 1 + \cot^{2} x \). Therefore, substituting this into our expression gives us: \[ 1 - \csc^{2} x = 1 - (1 + \cot^{2} x) = -\cot^{2} x. \] So the correct choice is \( -\cot^{2} x \). If you're ever stuck with trigonometric identities, try rewriting them using their corresponding reciprocal or Pythagorean properties. It’s like adjusting the puzzle pieces until they fit perfectly – a little trial and error can often lead you to the answer! And don’t forget that the identities are just like secret keys to unlock the treasure of trigonometric functions! Memorizing a few basic identities can go a long way in solving complex problems, transforming those brain-busters into a smooth stroll through math land.