Which expression is equivalent to (secβ−1)(secβ+1)/sec^2β for all values of β for which (secβ−1)(secβ+1)/sec^2β is defined?

Question

which is the correct answer?

secβtanβ

sin2β

secβ

tanβ

Answer 1

(secß-1)(secß+1)/sec²ß

The numerator is factors of difference of two squares.

=(sec²β−1)/sec²β

One of the Pythagorean identities, tan²β=sec²β−1

=tan²β/sec²β

Values of tangent and secant in terms of sine and cosine:

=(sin²β/cos²β)/(1/cos²β)

Use the rule for division of fractions (keep, change, flip)

=(sin²β/cos²β)*cos²β

Cancel cos²β

=sin²β (final answer)

Answer 2

Evaluate the numerator first (sec -1)(sec+1) = sec^2-1 which equals tan^2= Sin^2/Cos^2

Evaluate the denominator Sec^2 = 1/Cos^2 So the fraction becomes

sin^2/Cos^2. / 1/Cos^2= sin^2 so your answer is choice 2. Sin^2 B

Answer 3

If the question is what is (sec²ß-1)/(sec²ß), then the answer is 1-cos²ß=sin²ß