Simplifying Trig fractions

[math.]

Problem: \(\displaystyle \dfrac{sin(a) - sec(a)}{csc(a) - cos(a)}\)

I've gotten it so that it's at \(\displaystyle \dfrac{sin(a) - (1/cos(a))}{(1/sin(a)) - cos(a)}\)

 

[DrPhil]

Problem: \(\displaystyle \dfrac{sin(a) - sec(a)}{csc(a) - cos(a)}\)

I've gotten it so that it's at \(\displaystyle \dfrac{sin(a) - (1/cos(a))}{(1/sin(a)) - cos(a)}\)

Fractions are often a bit of a mess, so I recomment that you combine the two terms in the numerator over the common denominator of cos(a),
AND also combine the two terms if the denominator over the common denominator of sin(a).

Well - that turns out to be interesting...

 

[princeps]

Problem: \(\displaystyle \dfrac{sin(a) - sec(a)}{csc(a) - cos(a)}\)

I've gotten it so that it's at \(\displaystyle \dfrac{sin(a) - (1/cos(a))}{(1/sin(a)) - cos(a)}\)

\(\displaystyle \frac{\sin(a) - (1/\cos(a))}{(1/\sin(a)) - \cos(a)}=\frac{(\sin(a)\cos(a)-1)/\cos(a)}{(1-\sin(a)\cos(a))/\sin(a)}=-\frac{\sin(a)}{\cos(a)}=-\tan(a)\)