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. cos α - sec α choose the correct answer below. a. cos² α - 1 b. - sin α tan α c. - cos² α cot² α d. cos² α e. sin² α f. 1

write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression. cos α - sec α choose the correct answer below. a. cos² α - 1 b. - sin α tan α c. - cos² α cot² α d. cos² α e. sin² α f. 1

Answer

Explanation:

Step1: Substitute (\sec\alpha) with (\frac{1}{\cos\alpha})

(\cos\alpha-\sec\alpha=\cos\alpha - \frac{1}{\cos\alpha})

Step2: Get a common denominator

(\cos\alpha - \frac{1}{\cos\alpha}=\frac{\cos^{2}\alpha - 1}{\cos\alpha})

Step3: Use the Pythagorean identity (\sin^{2}\alpha+\cos^{2}\alpha = 1), so (\cos^{2}\alpha-1=-\sin^{2}\alpha)

(\frac{\cos^{2}\alpha - 1}{\cos\alpha}=\frac{-\sin^{2}\alpha}{\cos\alpha})

Step4: Rewrite (\frac{\sin\alpha}{\cos\alpha}) as (\tan\alpha)

(\frac{-\sin^{2}\alpha}{\cos\alpha}=-\sin\alpha\times\frac{\sin\alpha}{\cos\alpha}=-\sin\alpha\tan\alpha)

Answer:

B. (-\sin\alpha\tan\alpha)