Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?$
A) $1 + \cot \theta$
B) $\sec \theta \csc \theta$
C) $\sin \theta \tan \theta$
D) $- 2 \tan ^ { 2 } \theta$

Question

Quizplus · Accepted Answer

The identity can be simplified using trigonometric identities. $\tan^2\theta = \sec^2\theta - 1$ and $\tan\theta\sec\theta = \sin\theta$. So, $\tan^2\theta - 3\sin\theta\tan\theta\sec\theta = \sec^2\theta - 1 - 3\sin^2\theta$. Since $\sin^2\theta = 1 - \cos^2\theta$ and $\sec^2\theta = 1 + \tan^2\theta$, substituting these into the equation doesn't directly lead to any of the given options simplifying directly to them. However, the correct approach involves recognizing that the original manipulation was incorrect and that directly simplifying the given expression doesn't neatly match the provided options without further manipulation. The correct answer, based on re-evaluating the expression and the trigonometric identities involved, should focus on simplifying or transforming the given expression in a way that might directly relate to the identities or combinations thereof. However, without the correct manipulation leading to a straightforward answer, the explanation provided initially misleads to an incorrect conclusion. The correct simplification process should involve recognizing the relationships between the trigonometric functions involved and correctly applying trigonometric identities to simplify or transform the given expression to match one of the provided options.

Choose the One Alternative That Best Completes the Statement or Answers