Question

Express sec t + tan t in terms of sin t. Assume t is in the 2nd quadrant

Answer 1

\[\frac{1}{\cos(t)}+\frac{\sin(t)}{\cos(t)}=\frac{1+\sin(t)}{\cos(t)}\]
remember \[\sin^2(t)+\cos^2(t)=1 => \cos^2(t)=1-\sin^2(t) => \cos(t)= \pm \sqrt{1-\sin^2(t)}\]
since we are in the second quadrant then cos(t) is negative
\[\cos(t)=-\sqrt{1-\sin^2(t)}\]

Answer 2

\[\frac{1+\sin(t)}{-\sqrt{1-\sin^2(t)}}\]