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?

Answer 1

Answer: Choice E)  1

Work Shown:

$\frac{(1-\sin(\beta))(1+\sin(\beta))}{\cos^2(\beta)}\\\\\frac{1-\sin^2(\beta)}{\cos^2(\beta)}\\\\\frac{\cos^2(\beta)}{\cos^2(\beta)}\\\\1$

Therefore, $\frac{(1-\sin(\beta))(1+\sin(\beta))}{\cos^2(\beta)}=1$ 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.