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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

Sagot :

Answer: Choice E)  1

Work Shown:

[tex]\frac{(1-\sin(\beta))(1+\sin(\beta))}{\cos^2(\beta)}\\\\\frac{1-\sin^2(\beta)}{\cos^2(\beta)}\\\\\frac{\cos^2(\beta)}{\cos^2(\beta)}\\\\1\\\\[/tex]

Therefore, [tex]\frac{(1-\sin(\beta))(1+\sin(\beta))}{\cos^2(\beta)}=1\\\\[/tex] 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.

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