write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final…
write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression\n(1 + cot θ)(1 - tan θ)\nchoose the correct answer below.\na. cot θ - tan θ\nb. 1\nc. 2\nd. 0\ne. csc θ\nf. tan θ - cot θ
Answer
Explanation:
Step1: Substitute trigonometric identities
We know that (\cot\theta=\frac{\cos\theta}{\sin\theta}) and (\tan\theta = \frac{\sin\theta}{\cos\theta}). So ((1 + \cot\theta)(1-\tan\theta)=\left(1+\frac{\cos\theta}{\sin\theta}\right)\left(1-\frac{\sin\theta}{\cos\theta}\right)).
Step2: Simplify the expression
[ \begin{align*} \left(1+\frac{\cos\theta}{\sin\theta}\right)\left(1-\frac{\sin\theta}{\cos\theta}\right)&=\frac{\sin\theta+\cos\theta}{\sin\theta}\cdot\frac{\cos\theta - \sin\theta}{\cos\theta}\ &=\frac{\cos^{2}\theta-\sin^{2}\theta}{\sin\theta\cos\theta}\ &=\frac{\cos^{2}\theta}{\sin\theta\cos\theta}-\frac{\sin^{2}\theta}{\sin\theta\cos\theta}\ &=\frac{\cos\theta}{\sin\theta}-\frac{\sin\theta}{\cos\theta}\ &=\cot\theta - \tan\theta \end{align*} ]
Answer:
A. (\cot\theta-\tan\theta)