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

Question

Quizplus · Accepted Answer

The identity can be simplified using trigonometric identities. $\sin(2\theta) \tan(\theta) + \cos(2\theta)$ can be rewritten using the double angle formula for sine and the definition of tangent. The double angle formula for sine is $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, and $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Substituting these into the expression gives $2\sin(\theta)\cos(\theta) \cdot \frac{\sin(\theta)}{\cos(\theta)} + \cos(2\theta)$, which simplifies to $2\sin^2(\theta) + \cos(2\theta)$. Using the identity $\cos(2\theta) = 1 - 2\sin^2(\theta)$, we can substitute for $\cos(2\theta)$ to get $2\sin^2(\theta) + (1 - 2\sin^2(\theta))$, which simplifies to 1.

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