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Answer:
1 + sin(y).
Step-by-step explanation:
[tex] \frac{ \cos(y) }{ \sec(y) - \tan(y) } [/tex]
First, we rewrite sec(y) and tan(y) in terms of sin(y) and cos(y) by using the reciprocal identity for secant and the quotient identity for tangent:
[tex] \frac{ \cos(y) }{ \frac{1}{ \cos(y) } - \frac{ \sin(y) }{ \cos(y) } } [/tex]
Next, simplify the denominator:
[tex] \frac{ \cos(y) }{ \frac{1 - \sin(y) }{ \cos(y) } } [/tex]
Then, simplify the complex fraction:
[tex] \frac{ {( \cos(y) )}^{2} }{1 - \sin(y) } [/tex]
For cos²(y), use the Pythagorean identity and rewrite using cos²(y) = 1 - sin²(y):
[tex] \frac{1 - { (\sin(y)) }^{2} }{1 - \sin(y) } [/tex]
Notice that the numerator, 1 - sin²(y) is a difference of two squares. So, we rewrite it as (1 + sin(y))(1 - sin(y)):
[tex] \frac{(1 + \sin(y) )(1 - \sin(y) )}{1 - \sin(y) } [/tex]
Then cancel 1 - sin(y) from both the numerator and denominator. So, we are left with
[tex]1 + \sin(y) [/tex]
So, the answer is 1 + sin(y).