Question

question 5
simplify the following trigonometric expression.
\\(\dfrac{\cos^2 \alpha}{1 - \sin \alpha} - 1\\)
\\(\circ\\) 1
\\(\circ\\) \\(\sin \alpha\\)
\\(\circ\\) \\(\sin^2 \alpha\\)
\\(\circ\\) \\(\cos \alpha\\)

Explanation:

Step1: Use Pythagorean Identity

Recall the Pythagorean identity \( \cos^2\alpha = 1 - \sin^2\alpha \), which factors as \( (1 - \sin\alpha)(1 + \sin\alpha) \). Substitute this into the numerator of the first fraction:
\( \frac{(1 - \sin\alpha)(1 + \sin\alpha)}{1 - \sin\alpha} - 1 \)

Step2: Simplify the Fraction

Cancel the common factor \( 1 - \sin\alpha \) (assuming \( 1 - \sin\alpha
eq 0 \)):
\( (1 + \sin\alpha) - 1 \)

Step3: Combine Like Terms

Subtract 1 from \( 1 + \sin\alpha \):
\( 1 + \sin\alpha - 1 = \sin\alpha \)

Answer:

\(\sin\alpha\) (corresponding to the option "B. \(\sin\alpha\)" if options are labeled as A. 1, B. \(\sin\alpha\), C. \(\sin^2\alpha\), D. \(\cos\alpha\))