Question

Which expression is equivalent to (1−sinβ)(1+sinβ)/cos2β for all values of β for which (1−sinβ)(1+sinβ)/cos2β is defined?

Select the correct answer below:

tan2β
tanβ+secβ
tanβ
sec2β
1

Answer 1

Answer: Choice E) 1

Work Shown:

`((1-\sin(\beta))(1+\sin(\beta)))/(\cos^2(\beta))\\\\(1-\sin^2(\beta))/(\cos^2(\beta))\\\\(\cos^2(\beta))/(\cos^2(\beta))\\\\1\\\\`

Therefore,
`((1-\sin(\beta))(1+\sin(\beta)))/(\cos^2(\beta))=1\\\\` is an identity for all values of β for which the original expression is defined. I used the difference of squares rule for the 2nd step. Then I applied the pythagorean trig identity for the 3rd step.