Simplify: \(\frac{\cos 2\theta - 1}{\sin 2\theta}\)
FURTHER MATHEMATICS
WAEC 2012
Simplify: \(\frac{\cos 2\theta - 1}{\sin 2\theta}\)
- A. \(-\tan \theta\)
- B. \(-\cos \theta\)
- C. \(\tan \theta\)
- D. \(\cos \theta\)
Correct Answer: A. \(-\tan \theta\)
Explanation
\(\frac{\cos 2\theta - 1}{\sin 2\theta}\)
\(\cos (x + y) = \cos x \cos y - \sin x \sin y \implies \cos 2\theta = \cos^{2} \theta - \sin^{2} \theta\)
\(\cos^{2} \theta = 1 - \sin^{2} \theta \implies \cos 2\theta = 1 - 2\sin^{2} \theta\)
\(\sin 2\theta = 2\sin \theta \cos \theta\)
\(\therefore \frac{\cos 2\theta - 1}{\sin 2\theta} = \frac{1 - 2\sin^{2}\theta - 1}{2\sin \theta \cos \theta}\)
= \(\frac{-2 \sin^{2} \theta}{2\sin \theta \cos \theta} = \frac{- \sin \theta}{\cos \theta}\)
= \(-\tan \theta\)
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