for question (a):
(sec A - tan A)^2
expand first using (a
+ b)^2 = a^2 - 2ab + b^2.
= (sec A)^2 - 2 sec A tan A + (tan A)^2
expand
further.
= (1 / cos^2 A) - [2 (1 / cos A)(sin A / cosA) ] + ( sin^2
A / cos^2 A)
expand further and try to
simplify.
= (1 - 2 sin A + sin^2 A) / (cos^2 A)
this is what you get after
simplifying the above chunk of stuffs. remember sin^2 + cos^2 =1 and
hence, change the denominator.
= (1 - 2 sin A + sin^2 A) / (1 - sin^2 A)
change the
numerator from a^2 - 2ab + b^2 form to (a + b)^2 form. change the
denominator from a^2 - b^2 form to (a + b)(a
- b).
= (1 - sin A)^2 / [ (1 + sin A)(1 - sin A) ]
you can choose to open up the
numerator like what i've done below. this is optional
lah.
= [ (1 - sin A)(1 - sin A) ] / [ ( 1 + sin A)(1 - sin A) ]
cancel the common factor which
is (1 - sin A) in this case.
= 1 - sin A / 1 + sin A
(proven)
hope this helps. =)