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

Using a trigonometric identity, it is found that the equivalent expression is given by 1.

What are the trigonometric identities used to solve this question?

Relating sine and cosine, we have that:

$\sin^{2}{\beta} + \cos^{2}{\beta} = 1$

Then:

$\cos^{2}{\beta} = 1 - \sin^{2}{\beta}$

For the tangent, we have that:

$\tan{\beta} = \frac{\sin{\beta}}{\cos{\beta}}$.

For the secant, we have that:

$\sec{\beta} = \frac{1}{\cos{\beta}}$.

In this problem, the expression is:

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

More can be learned about trigonometric identities at brainly.com/question/7331447

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